#7616872

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1370 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)(9,10,12,14)(11,15,16,13)(17,18,23,22)(19,24,28,29)(20,25,30,31)(21,27,32,26);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,27)(21,31)(22,29)(23,28)(25,32)(26,30);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20)(18,25)(19,27)(21,29)(22,31)(23,30)(24,32)(26,28);(10,14)(13,15)(17,21,18,26,23,32,22,27)(19,20,29,25,28,30,24,31);(2,5)(6,8)(10,14)(13,15)(20,30)(21,32)(25,31)(26,27)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=fg,d^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7524068);