#7561489

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	#45385	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2160 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)(14,19)(20,32)(21,33)(22,30)(26,36)(34,43)(35,41)(37,44);(2,3)(6,10)(11,12)(13,14,17,19)(15,20,24,32)(16,26,27,36)(18,30,29,22)(21,31,33,28)(23,37,38,44)(25,41,40,35)(34,42,43,39);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,15,17,24)(14,20,19,32)(16,23,27,38)(18,25,29,40)(21,34,33,43)(22,35,30,41)(26,37,36,44)(28,39,31,42);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,16)(14,21)(15,23)(17,27)(18,31)(19,33)(20,34)(22,36)(24,38)(25,42)(26,30)(28,29)(32,43)(35,44)(37,41)(39,40);(6,10)(11,12)(15,24)(16,27)(20,32)(21,33)(25,40)(26,36)(28,31)(35,41)

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=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=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, 7468685);