#187290

order	512
nilpotency class	3
isoclinism	isoclinic class of #187213
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	#62835	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2161 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=h,e^2=i,g^2=h,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,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, 94486);