#7160941

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1635 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=gi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 7068137);