#6465402

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1426 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)(3,7)(4,6)(5,8)(17,18,24,21)(19,26,30,31)(20,28,29,25)(22,27,23,32);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,19,22,20,24,30,23,29)(18,25,32,31,21,28,27,26);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,20,23,19,24,29,22,30)(18,26,27,28,21,31,32,25);(9,11,12,16)(10,14,13,15)(17,21,24,18)(19,28,30,25)(20,26,29,31)(22,27,23,32);(1,4,5,3)(2,6,8,7)(17,22,24,23)(18,27,21,32)(19,29,30,20)(25,31,28,26)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=hi,e^2=fg,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,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372598);