#6631325

order	512
nilpotency class	3
isoclinism	isoclinic class of #6631114
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1574 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538521);