#105620

order	512
nilpotency class	3
isoclinism	isoclinic class of #105616
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

811 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)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18)(19,28)(20,32,34,25)(21,36)(22,26)(23,27)(24,30)(29,43,45,42)(31,40)(33,41,35,47)(37,39)(38,44,46,48);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(17,19,22,30,23,31,21,37)(18,24,26,40,27,39,36,28)(20,29,35,44,34,45,33,48)(25,38,41,43,32,46,47,42);(17,20)(18,25)(19,29)(21,33)(22,35)(23,34)(24,38)(26,41)(27,32)(28,42)(30,44)(31,45)(36,47)(37,48)(39,46)(40,43)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=dh,f^b=fi,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12816);