#7143320

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994043
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	#39852	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1697 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050516);