#126899

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=h,h^2=i,b^a=bd,d^a=dg,c^b=ci,e^b=eg

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, 34095);