#187143

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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2098 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,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, 94339);