#6370146

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂⁴
commutator	C₂²	C₂³×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂⁴×C₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2280 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:

(5,6)(7,10)(8,9)(11,12)(13,14,17,22,18,21,28,34)(15,19,25,31,23,29,36,40)(16,20,26,33,27,32,38,42)(24,30,37,41,35,39,43,44);(1,2,4,3)(6,9)(10,12)(13,15)(14,19)(16,24)(17,25)(18,23)(20,30)(21,29)(22,31)(26,37)(27,35)(28,36)(32,39)(33,41)(34,40)(38,43)(42,44);(1,3,4,2)(15,23)(19,29)(24,35)(25,36)(30,39)(31,40)(37,43)(41,44);(13,16,17,26,18,27,28,38)(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=g,b^2=h,c^2=h,d^2=g,e^2=f,g^2=i,b^a=bf,e^a=ei,c^b=ci

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277342);