#6990270

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885065
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1535 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)(3,7)(4,6)(5,8)(18,23)(19,26)(20,22)(24,31)(27,28)(30,32);(1,3,4,8)(2,5,6,7)(9,10,13,15)(11,14,12,16)(17,18)(19,24)(20,29)(21,23)(22,25)(26,32)(27,31)(28,30);(3,8)(5,7)(9,10)(11,16)(12,14)(13,15)(17,19,21,28)(18,24,23,30)(20,26,22,27)(25,31,29,32);(2,6)(5,7)(9,11,13,12)(10,14,15,16)(17,20,21,22)(18,25,23,29)(19,27,28,26)(24,32,30,31);(9,12,13,11)(10,14,15,16)(18,23)(24,30)(25,29)(31,32)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi

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, 6897466);