#187854

order	512
nilpotency class	3
isoclinism	isoclinic class of #187780
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3544	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2129 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=h,e^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=ei,g^b=ghi,d^c=dh,g^c=gi

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, 95050);