#382604

order	512
nilpotency class	3
isoclinism	isoclinic class of #382477
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2310
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14339
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1137 conjugacy classes of subgroups; only the 312 normal subgroups are shown.

normal subgroups only

This group has 1137 conjugacy classes of subgroups — too many to draw. Only its 312 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289800);