#7561329

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#35737	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2165 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)(7,11)(8,12)(9,10)(13,14)(15,23)(16,20)(17,21)(18,22)(19,26)(24,37)(25,33)(27,32)(28,34)(29,35)(30,36)(31,39)(38,43)(40,42)(41,44);(2,3)(6,10)(11,12)(13,15,17,26)(14,19,21,23)(16,24,28,39)(18,25,30,27)(20,31,34,37)(22,32,36,33)(29,38,41,40)(35,42,44,43);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,17,28)(14,20,21,34)(15,24,26,39)(18,29,30,41)(19,31,23,37)(22,35,36,44)(25,38,27,40)(32,42,33,43);(1,3,4,2)(5,8,9,7)(6,12,10,11)(15,25)(18,30)(19,32)(22,36)(23,33)(24,38)(26,27)(29,41)(31,42)(35,44)(37,43)(39,40);(6,10)(11,12)(14,21)(16,28)(19,23)(22,36)(24,39)(29,41)(32,33)(38,40)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=fgi,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,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7468525);