#6570017

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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	#2408	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1450 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=g,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313384, #72313494, #72313904
maximal normal subgroup of	#72313384, #72313494, #72313904

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6477213);