#107674

order	512
nilpotency class	3
isoclinism	isoclinic class of #107658
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

787 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:

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

Polycyclic presentation:

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

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, 14870);