#500808

order	512
nilpotency class	4
isoclinism	isoclinic class of #500807
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24260
commutator, Frattini, cosocle	C₂×#41	C₂⁴
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1017 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1017 conjugacy classes of subgroups — too many to draw. Only its 138 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)(3,15)(4,10)(5,9)(6,27)(7,29)(8,14)(11,19)(12,16)(13,32)(17,45)(18,41)(20,52)(21,33)(22,55)(23,39)(24,40)(25,43)(26,35)(28,42)(30,46)(31,44)(34,58)(36,53)(37,50)(38,47)(48,59)(49,64)(51,63)(54,61)(56,60)(57,62);(1,3,16,26,8,19,6,5)(2,9,27,11,14,35,12,15)(4,20,47,57,24,51,22,53)(7,30,48,25,32,18,28,58)(10,36,55,63,40,62,38,52)(13,43,59,46,29,34,42,41)(17,49,54,31,50,56,21,23)(33,60,37,44,61,64,45,39);(1,4)(2,10)(3,17)(5,21)(6,22)(7,31)(8,24)(9,33)(11,37)(12,38)(13,44)(14,40)(15,45)(16,47)(18,51)(19,50)(20,30)(23,32)(25,57)(26,54)(27,55)(28,49)(29,39)(34,62)(35,61)(36,43)(41,52)(42,60)(46,63)(48,56)(53,58)(59,64);(1,5,6,19,8,26,16,3)(2,11,12,9,14,15,27,35)(4,21,22,50,24,54,47,17)(7,25,28,30,32,58,48,18)(10,37,38,33,40,45,55,61)(13,41,42,34,29,46,59,43)(20,23,53,56,51,31,57,49)(36,44,52,60,62,39,63,64)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312431, #72312462, #72312507, …
maximal normal subgroup of	#72312431, #72312462, #72312507, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408004);