#7611903

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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

2005 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)(15,21)(16,22)(17,32)(18,24)(19,23)(20,27)(25,37)(26,35)(28,42)(29,44)(30,40)(31,39)(33,36)(34,41)(38,43);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,28)(17,26)(18,27)(19,33)(21,24)(22,38)(23,35)(25,31)(29,41)(30,43)(32,36)(34,39)(37,44)(40,42);(6,10)(7,11)(13,15)(14,21)(16,25)(17,33)(18,27)(19,26)(20,24)(22,37)(23,35)(28,31)(29,43)(30,41)(32,36)(34,40)(38,44)(39,42);(6,10)(9,12)(13,16,17,29,18,30,19,31)(14,22,32,44,24,40,23,39)(15,25,33,43,27,41,26,28)(20,34,35,42,21,37,36,38);(1,3,4,2)(13,17,18,19)(14,23,24,32)(15,26,27,33)(16,29,30,31)(20,35,21,36)(22,39,40,44)(25,28,41,43)(34,42,37,38)

Polycyclic presentation:

2^9|b^2=fi,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,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, 7519099);