#106386

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	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

882 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,f^2=h,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, 13582);