#106391

order	512
nilpotency class	3
isoclinism	isoclinic class of #106380
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

794 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)(3,10,12,16)(4,8)(5,14)(6,9)(7,11,15,13)(17,18,21,27,23,28,22,26)(19,29,33,38,32,39,31,24)(20,25,35,40,36,42,34,41)(30,44,45,37,46,43,47,48);(1,3,5,13,6,12,4,11)(2,7,8,10,9,15,14,16)(17,19,22,31,23,32,21,33)(18,24,26,39,28,38,27,29)(20,30,34,47,36,46,35,45)(25,37,41,44,42,48,40,43);(17,20)(18,25)(19,30)(21,35)(22,34)(23,36)(24,37)(26,41)(27,40)(28,42)(29,43)(31,47)(32,46)(33,45)(38,48)(39,44)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13587);