#7062362

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45092	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1870 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,23,19,24,15,20)(16,27,25,36,29,38,33,21)(17,30,31,34,32,22,26,37)(28,41,39,35,43,40,42,44);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15,19,18)(14,20,24,23)(16,25,29,33)(17,26,32,31)(21,27,36,38)(22,34,30,37)(28,39,43,42)(35,40,44,41);(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,25)(17,28)(18,33)(19,29)(20,27)(22,35)(23,38)(24,36)(26,39)(30,44)(31,42)(32,43)(34,40)(37,41);(2,4)(7,12)(9,11)(13,17,19,32)(14,22,24,30)(15,26,18,31)(16,28,29,43)(20,34,23,37)(21,35,36,44)(25,39,33,42)(27,40,38,41);(6,10)(9,11)(13,15,19,18)(14,20,24,23)(16,25,29,33)(17,31,32,26)(21,27,36,38)(22,37,30,34)(28,42,43,39)(35,41,44,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6969558);