#6727915

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1239 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)(3,6)(4,7)(5,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26,30,35)(27,29,34,32)(28,39,31,33)(36,38,40,37);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,26)(27,36)(28,33)(29,37)(30,35)(31,39)(32,38)(34,40);(1,3,5,4)(2,7,8,6)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27)(26,32)(28,38)(29,35)(30,34)(31,37)(33,40)(36,39);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,28,30,31)(26,33,35,39)(27,37,34,38)(29,40,32,36);(1,4,5,3)(2,7,8,6)(11,21)(15,18)(19,20)(23,24)(25,29)(26,34)(27,35)(28,40)(30,32)(31,36)(33,38)(37,39)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fg,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6635111);