#187141

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	#62815	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2168 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:

(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(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);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(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);(2,7)(4,13)(6,9)(10,12)(17,20)(18,25)(19,30)(21,34)(22,37)(23,35)(24,38)(26,40)(27,42)(28,33)(29,43)(31,44)(32,46)(36,45)(39,47)(41,48);(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,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, 94337);