#7488870

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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	#42688	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1699 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7396066);