#499904

order	512
nilpotency class	4
isoclinism	isoclinic class of #499904
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3192
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23918
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1379 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1379 conjugacy classes of subgroups — too many to draw. Only its 174 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)(5,21)(6,26)(7,11)(9,19)(10,25)(12,24)(13,38)(14,29)(15,44)(16,47)(18,50)(20,52)(22,37)(23,53)(27,35)(28,40)(30,46)(31,43)(32,60)(33,51)(34,54)(36,55)(39,49)(41,61)(42,62)(45,59)(48,58)(56,63)(57,64);(1,3)(2,8)(4,13)(5,15)(6,16)(7,14)(9,32)(10,30)(11,29)(12,31)(17,49)(18,39)(19,40)(20,41)(21,44)(22,45)(23,42)(24,43)(25,46)(26,47)(27,48)(28,33)(34,59)(35,61)(36,57)(37,56)(38,50)(51,60)(52,58)(53,62)(54,63)(55,64);(1,4)(2,9)(3,13)(5,22)(6,27)(7,18)(8,28)(10,35)(11,33)(12,37)(14,39)(15,45)(16,48)(17,21)(19,25)(20,24)(23,54)(26,34)(29,32)(30,58)(31,59)(36,52)(38,44)(40,46)(41,43)(42,63)(47,56)(49,62)(50,53)(51,55)(57,61)(60,64);(3,14)(5,23)(8,29)(10,36)(13,39)(16,43)(17,50)(19,51)(21,53)(22,54)(25,55)(28,32)(31,47)(35,52)(41,48)(56,59)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307496, #72307945, #72307958
maximal normal subgroup of	#72307496, #72307945, #72307958

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407100);