#6883822

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1748 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|c^2=fgi,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257286, #72257462, #72257608
maximal normal subgroup of	#72257286, #72257462, #72257608
maximal quotient of	#72265647

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6791018);