#187293

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	#62828	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2053 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)(3,10,12,15)(4,13)(5,8)(6,9)(7,11,16,14)(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)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,24)(20,29)(21,32)(22,30)(23,31)(25,38)(26,42)(27,28)(33,43)(34,46)(35,45)(36,39)(37,44)(40,48)(41,47);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(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);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(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,d^2=hi,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, 94489);