#107874

order	512
nilpotency class	3
isoclinism	isoclinic class of #107858
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

810 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)(3,6,10,16)(4,9)(5,8)(7,11,15,12)(13,14)(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)(2,6,8,16)(4,12)(5,10)(7,14,15,9)(11,13)(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);(1,3)(2,7)(4,12)(5,10)(6,14)(8,15)(9,16)(11,13)(17,20)(18,25)(19,29)(21,36)(22,33)(23,34)(24,37)(26,40)(27,41)(28,42)(30,47)(31,46)(32,45)(35,39)(38,48)(43,44)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=di,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=fi

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, 15070);