#187137

order	512
nilpotency class	3
isoclinism	isoclinic class of #186994
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×#3512	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2174 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:

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

Polycyclic presentation:

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

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, 94333);