#6373259

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#34636	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2198 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ci,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280455);