#105405

order	512
nilpotency class	3
isoclinism	isoclinic class of #7943
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#95
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄×#95
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

607 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)(4,12)(5,8)(6,11)(7,13)(10,15)(14,16)(17,18)(19,27,29,23)(20,24)(21,26)(22,25)(28,37,31,35)(30,41,43,36)(32,39)(33,38)(34,40)(42,47,44,46)(45,48);(1,3)(2,6,8,14)(4,11)(5,10)(7,9,12,15)(13,16)(17,19)(18,23)(20,31)(21,30)(22,29)(24,37)(25,27)(26,36)(28,33)(32,44)(34,43)(35,38)(39,47)(40,41)(42,45)(46,48);(3,10)(6,14)(9,15)(11,16)(17,20,21,32,22,33,34,45)(18,24,26,39,25,38,40,48)(19,28,30,42,29,31,43,44)(23,35,36,46,27,37,41,47)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=hi,g^2=i,b^a=bd,d^a=dh,f^a=fhi,c^b=ch,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12601);