#127168

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

415 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)(3,11,13,7)(4,8)(5,10)(6,9)(12,16,14,15)(17,18,20,25,22,27,34,41,23,28,21,26,33,40,35,42)(19,24,30,37,32,39,44,47,29,36,31,38,43,46,45,48);(1,3,5,14,6,13,4,12)(2,7,10,16,9,11,8,15)(17,19,21,31,22,32,35,45,23,29,20,30,33,43,34,44)(18,24,26,38,27,39,42,48,28,36,25,37,40,46,41,47);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(19,29)(24,36)(30,31)(32,43)(37,38)(39,46)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,e^2=h,f^2=gh,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, 34364);