#6461573

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23421	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6368769);