#6548245

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	#38216	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1606 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)(17,18,22,28,21,25,20,24)(19,23,29,32,26,30,27,31);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,27)(21,26)(22,29)(24,31)(25,30)(28,32);(10,13)(14,16)(17,18,21,25)(19,23,26,30)(20,28,22,24)(27,32,29,31);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(19,26)(23,30)(27,29)(31,32);(11,15)(14,16)(19,26)(23,30)(27,29)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6455441);