#126970

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

484 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:

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

Polycyclic presentation:

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

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, 34166);