#187779

order	512
nilpotency class	3
isoclinism	isoclinic class of #187766
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	C₂×#3512	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2215 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)(3,9,11,6)(4,7)(5,8)(10,15,12,14)(13,16)(17,18)(19,27)(20,31,33,24)(21,34)(22,26)(23,30)(25,36)(28,42,43,41)(29,40)(32,39,48,46)(35,38)(37,44,45,47);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,22,29)(18,23,26,38)(20,28,33,43)(21,35,36,30)(24,37,31,45)(25,40,34,27)(32,47,48,44)(39,42,46,41);(17,20)(18,24)(19,28)(21,32)(22,33)(23,37)(25,39)(26,31)(27,41)(29,43)(30,44)(34,46)(35,47)(36,48)(38,45)(40,42);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(19,29)(23,38)(27,40)(28,43)(30,35)(37,45)(41,42)(44,47)

Polycyclic presentation:

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

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, 94975);