#7162118

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1693 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)(17,18,23,20)(19,26,30,27)(21,25,22,32)(24,28,31,29);(2,5)(3,7)(10,13)(11,15)(17,19,21,28,23,30,22,29)(18,24,32,27,20,31,25,26);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18,23,20)(19,24,30,31)(21,32,22,25)(26,28,27,29);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,23,18)(19,27,30,26)(21,32,22,25)(24,29,31,28);(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,25,20,32)(19,28,30,29)(24,26,31,27)

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=eh,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, 7069314);