#126803

order	512
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

504 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=gi,e^2=h,f^2=gh,h^2=i,b^a=bd,d^a=dg,f^a=fg,c^b=ci,d^b=di,e^b=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33999);