#187914

order	512
nilpotency class	3
isoclinism	isoclinic class of #187872
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2017 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)(3,13,14,4,9,15,16,8)(17,18,22,28,23,26,21,27)(19,29,32,39,33,24,31,40)(20,34,37,25,35,42,36,41)(30,44,47,43,45,48,46,38);(1,3)(2,8)(4,12)(5,14)(6,16)(7,9)(10,13)(11,15)(17,19,23,33)(18,24,26,29)(20,30,35,45)(21,32,22,31)(25,38,41,43)(27,40,28,39)(34,44,42,48)(36,47,37,46);(1,4)(2,9)(3,12)(5,15)(6,13)(7,8)(10,16)(11,14)(17,20)(18,25)(19,30)(21,37)(22,36)(23,35)(24,38)(26,41)(27,42)(28,34)(29,43)(31,47)(32,46)(33,45)(39,44)(40,48);(1,5,7,6)(2,10,12,11)(3,14,9,16)(4,15,8,13)(20,35)(25,41)(30,45)(34,42)(36,37)(38,43)(44,48)(46,47)

Polycyclic presentation:

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

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, 95110);