#6915213

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1564 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:

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

Polycyclic presentation:

2^9|c^2=fgi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308682, #72308996
maximal normal subgroup of	#72308682, #72308996

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6822409);