#6461439

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	#35328	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1988 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=ghi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,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, 6368635);