#127036

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	C₄²	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	#97
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

417 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,9,10,6)(4,8)(5,7)(11,15,16,13)(12,14)(17,18,21,26,22,27,36,44,23,28,37,45,20,25,34,42)(19,29,32,46,31,39,47,43,30,41,35,48,33,24,38,40);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,19)(18,24)(20,33)(21,35)(22,31)(23,30)(25,41)(26,43)(27,29)(28,39)(32,37)(34,47)(36,38)(40,45)(42,46)(44,48);(19,30)(24,39)(29,41)(31,33)(32,35)(38,47)(40,43)(46,48)

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=i,e^2=h,f^2=g,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, 34232);