#382735

order	512
nilpotency class	3
isoclinism	isoclinic class of #382440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2309
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14342
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1061 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

normal subgroups only

This group has 1061 conjugacy classes of subgroups — too many to draw. Only its 272 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,8)(4,17,19,46)(5,10)(6,11)(7,12)(9,20,31,48)(13,36,38,44)(14,26)(15,27)(16,28)(18,45,21,49)(22,33)(23,34)(24,35)(25,39,29,57)(30,47,32,50)(37,59,40,61)(41,54)(42,55)(43,56)(51,58)(52,60,53,62)(63,64);(1,3,6,15,7,16,24,43)(2,8,11,27,12,28,35,56)(4,13,20,39,19,38,48,57)(5,14,22,41,23,42,51,63)(9,29,17,44,31,25,46,36)(10,26,33,54,34,55,58,64)(18,37,47,60,21,40,50,62)(30,53,45,61,32,52,49,59);(1,4)(2,9,12,31)(3,13)(5,21,23,18)(6,20)(7,19)(8,25,28,29)(10,32)(11,17,35,46)(14,40,42,37)(15,39)(16,38)(22,50,51,47)(24,48)(26,53)(27,36,56,44)(30,34)(33,49)(41,62,63,60)(43,57)(45,58)(52,55)(54,61)(59,64);(1,5)(2,10)(3,14)(4,18)(6,22)(7,23)(8,26)(9,30)(11,33)(12,34)(13,37)(15,41)(16,42)(17,45)(19,21)(20,47)(24,51)(25,52)(27,54)(28,55)(29,53)(31,32)(35,58)(36,59)(38,40)(39,60)(43,63)(44,61)(46,49)(48,50)(56,64)(57,62)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,f^2=i,b^a=be,c^a=cf,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285720, #72286021, C₂×#382735
maximal normal subgroup of	#72285720, #72286021, C₂×#382735
maximal quotient of	C₂×#382735

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289931);