#6623526

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1540 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)(3,6)(4,5)(7,8)(10,13)(14,16)(17,18,22,26)(19,27,30,31)(20,23,21,25)(24,29,32,28);(2,5)(6,8)(10,13)(11,15)(18,23)(20,21)(24,31)(25,26)(27,32)(28,29);(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,26,32)(20,28,21,29)(23,31,25,27);(1,3,4,7)(2,6,5,8)(10,13)(11,15)(17,20,22,21)(18,23,26,25)(19,28,30,29)(24,31,32,27);(9,11,12,15)(10,14,13,16)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=fghi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6530722);