#7611944

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#45332	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2003 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7519140);