#7197858

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
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	#24150	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1526 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,4,6)(3,7,8,5)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,30,35,31,36,29,34)(27,37,28,33,38,40,39,32);(1,3,4,8)(2,5,6,7)(11,17)(12,20)(14,21)(15,19)(25,27)(26,32)(28,29)(30,39)(31,38)(33,36)(34,37)(35,40);(2,6)(5,7)(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24);(3,8)(5,7)(11,17)(14,21)(18,23)(22,24)(25,28,29,27,31,39,30,38)(26,33,34,37,36,32,35,40);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,29,31,30)(26,34,36,35)(27,28,38,39)(32,37,33,40)

Polycyclic presentation:

2^9|a^2=fh,b^2=g,c^2=h,d^2=fgi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 7105054);