#7621127

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
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	#26954	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#126 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1436 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,28)(20,31)(21,32)(22,27)(23,26)(24,30)(25,29);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,27,28)(20,21,29,23)(25,32,31,26);(1,3,5,4)(2,7,6,8)(10,13)(14,16);(10,13)(14,16)(17,20,23,19,22,29,21,30)(18,25,26,24,27,31,32,28);(1,4,5,3)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,26,27,32)(19,29,30,20)(24,25,28,31)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,d^2=gh,e^2=fg,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^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, 7528323);