#6457135

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1792 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)(9,10,12,14)(11,13,15,16)(17,18,23,27)(19,28,20,31)(21,25,24,32)(22,30,29,26);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,25,23,20,29,32)(18,24,26,28,27,21,30,31);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20,22,32,23,19,29,25)(18,21,26,31,27,24,30,28);(11,15)(13,16)(17,21)(18,25)(19,30)(20,26)(22,31)(23,24)(27,32)(28,29);(19,20)(21,24)(25,32)(28,31)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6364331);