#6474972

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470867
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1399 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)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,25)(19,27,24,23)(20,30,31,29)(22,28,32,26);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,26)(21,24)(22,29)(25,27)(28,31)(30,32);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,20,22,27,21,31,32,23)(18,24,26,30,25,19,28,29);(11,15)(13,16)(19,24)(20,31)(23,27)(29,30);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(18,25)(19,24)(26,28)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6382168);