#105384

order	512
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂×C₂²⋊C₄
commutator	C₂³	C₄³
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄×#95
prime radical	C₂⁴	C₂×C₄²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

628 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:

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

Polycyclic presentation:

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

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, 12580);