#187268

order	512
nilpotency class	3
isoclinism	isoclinic class of #187195
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	C₂×#3561	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2193 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94464);