#7328586

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324436
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1329 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,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, 7235782);