#7328588

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	#30609	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1451 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)(25,26,29,28)(27,33,35,40)(30,37,38,32)(31,39,34,36);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(26,28)(31,34)(32,37)(33,40);(1,3)(2,5)(4,8)(6,7)(25,27,30,36,29,35,38,39)(26,31,32,33,28,34,37,40);(2,6)(3,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28,29,26)(27,34,35,31)(30,37,38,32)(33,36,40,39);(2,6)(5,7)(10,16)(14,17)(15,24)(22,23)(26,28)(31,34)(32,37)(33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,d^2=f,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, 7235784);