#7200340

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2323 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)(9,10)(11,13)(12,14)(15,16)(17,18,20,23)(19,25,27,32)(21,30,29,24)(22,31,26,28);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,18)(19,26)(20,23)(21,24)(22,27)(25,31)(28,32)(29,30);(11,15)(13,16)(17,19,20,27)(18,22,23,26)(21,31,29,28)(24,32,30,25);(3,8)(5,7)(10,14)(11,15)(17,18)(19,22)(20,23)(21,24)(25,28)(26,27)(29,30)(31,32);(2,6)(5,7)(11,15)(13,16)(18,23)(22,26)(24,30)(25,32)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7107536);