#7206092

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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	#35715	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1556 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)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18,23,27)(19,24,22,31)(20,26,21,30)(25,29,32,28);(1,3)(2,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19,23,22)(18,24,27,31)(20,29,21,28)(25,26,32,30);(3,7)(5,8)(10,14)(15,16)(17,20,23,21)(18,25,27,32)(19,28,22,29)(24,30,31,26);(2,6)(3,7)(9,11,13,12)(10,15,14,16)(17,21,23,20)(18,26,27,30)(19,28,22,29)(24,32,31,25);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(20,21)(25,32)(26,30)(28,29)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,e^c=ei,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, 7113288);