#6916065

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
prime radical	#30714	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1461 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fg,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308665, #72309006
maximal normal subgroup of	#72308665, #72309006

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6823261);