#106336

order	512
nilpotency class	3
isoclinism	isoclinic class of #106326
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

817 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)(17,18,22,26,23,27,21,35)(19,28,32,38,31,43,30,24)(20,25,33,40,34,41,36,39)(29,44,45,37,46,42,47,48);(1,3,5,13,6,12,4,11)(2,7,8,10,9,15,14,16)(17,19)(18,24,27,38)(20,29)(21,30)(22,32)(23,31)(25,37,41,48)(26,28,35,43)(33,45)(34,46)(36,47)(39,44,40,42);(17,20,23,34)(18,25,27,41)(19,29,31,46)(21,36,22,33)(24,37,38,48)(26,40,35,39)(28,42,43,44)(30,47,32,45)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,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, 13532);