#7511788

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500695
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1239 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei

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, 7418984);