#126521

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

410 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,h^2=i,b^a=bg,c^a=ci

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33717);