#6537605

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	#24438	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1646 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6444801);