#6915219

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	#30398	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1431 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)(11,19)(12,13)(14,17)(15,18)(16,22)(20,23)(21,24)(25,26)(27,36)(28,38)(29,34)(30,35)(31,39)(32,33)(37,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,20)(13,21)(14,23)(15,22)(17,24)(18,19)(25,27,30,33)(26,32,35,36)(28,34,40,39)(29,38,31,37);(1,3)(2,5)(4,8)(6,7)(9,12,15,17)(10,13,18,14)(11,20,22,24)(16,21,19,23)(25,28,29,36,30,40,31,32)(26,33,39,37,35,27,34,38);(3,8)(5,7)(9,13,15,14)(10,17,18,12)(11,21,22,23)(16,24,19,20)(25,27)(26,32)(28,39)(29,38)(30,33)(31,37)(34,40)(35,36);(2,6)(5,7)(9,14,15,13)(10,17,18,12)(11,21,22,23)(16,20,19,24)(25,29,30,31)(26,34,35,39)(27,37,33,38)(28,36,40,32)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=fi,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6822415);