#186993

order	512
nilpotency class	3
isoclinism	isoclinic class of #186985
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62831	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2062 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94189);