#6550700

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	#24162	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1445 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,31,39,30,34,29,35)(27,36,37,40,28,32,38,33);(2,6)(7,8)(10,14)(11,17)(16,21)(18,23)(25,27,30,28)(26,32,34,36)(29,38,31,37)(33,35,40,39);(1,3,5,4)(2,7,6,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28,30,27)(26,33,34,40)(29,38,31,37)(32,35,36,39);(1,4,5,3)(2,8,6,7)(12,19)(16,21)(18,23)(22,24)(26,34)(32,36)(33,40)(35,39);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,30,31)(26,35,34,39)(27,37,28,38)(32,33,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313366, #72313476, #72313813
maximal normal subgroup of	#72313366, #72313476, #72313813

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6457896);