#493887

order	512
nilpotency class	4
isoclinism	isoclinic class of #480263
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁴×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1718 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,13,17,14,18,29,36)(11,19,22,37,20,33,27,39)(12,23,25,35,26,16,28,34)(15,30,32,24,31,38,40,21);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,24)(13,27)(14,20)(16,33)(17,32)(18,31)(19,23)(21,26)(22,29)(25,30)(28,38)(34,39)(35,37)(36,40);(11,20)(15,31)(19,33)(21,24)(22,27)(30,38)(32,40)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267418, #72267625
maximal normal subgroup of	#72267418, #72267625
maximal quotient of	#72267426, #72267459, #72267467

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401083);