#7611939

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	#45248	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)(15,22)(16,21)(17,30)(18,35)(19,25)(20,24)(23,33)(26,36)(27,37)(28,39)(29,38)(31,40)(32,41)(34,44)(42,43);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,21)(16,19)(17,31)(18,27)(20,29)(22,25)(23,40)(24,37)(26,34)(28,32)(30,43)(33,42)(35,38)(36,41)(39,44);(6,10)(7,11)(13,16,19,15)(14,22,25,21)(17,26,33,28)(18,27,20,29)(23,36,30,39)(24,37,35,38)(31,32,42,34)(40,44,43,41);(6,10)(9,12)(13,17,18,32,19,33,20,34)(14,23,35,44,25,30,24,41)(15,26,29,42,16,28,27,31)(21,36,37,43,22,39,38,40);(1,3,4,2)(13,18,19,20)(14,24,25,35)(15,27,16,29)(17,32,33,34)(21,37,22,38)(23,41,30,44)(26,31,28,42)(36,43,39,40)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=hi,e^2=f,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, 7519135);