#7200385

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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

2361 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7107581);