#7620666

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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×#496	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1375 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fi,d^2=fghi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7527862);