#107399

order	512
nilpotency class	3
isoclinism	isoclinic class of #107386
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

791 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)(3,10,12,6)(4,7)(5,8)(9,13)(11,15,16,14)(17,18,21,26,23,27,22,36)(19,28,31,42,30,24,32,38)(20,25,34,40,33,39,35,46)(29,37,44,47,43,41,45,48);(1,3)(2,6)(4,11)(5,12)(7,14)(8,10)(9,15)(13,16)(17,19,22,32,23,30,21,31)(18,24,26,38,27,28,36,42)(20,29,35,45,33,43,34,44)(25,37,40,47,39,41,46,48);(2,7)(4,13)(6,14)(8,9)(10,15)(11,16)(17,20)(18,25)(19,29)(21,34)(22,35)(23,33)(24,37)(26,40)(27,39)(28,41)(30,43)(31,44)(32,45)(36,46)(38,47)(42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14595);