#6370915

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4386 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

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

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, 6278111);