#7162182

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=i,d^2=gi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi

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, 7069378);