#6460648

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1822 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=fgh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,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, 6367844);