#187269

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	#62748	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2199 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18,22,26,23,27,21,35)(19,28,30,39,31,24,36,40)(20,32,33,41,34,25,37,42)(29,43,46,48,45,38,44,47);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19)(18,24)(20,29)(21,30)(22,36)(23,31)(25,38)(26,39)(27,28)(32,43)(33,44)(34,45)(35,40)(37,46)(41,47)(42,48);(4,13)(8,15)(9,16)(10,12)(17,20)(18,25)(19,29)(21,33)(22,37)(23,34)(24,38)(26,41)(27,32)(28,43)(30,44)(31,45)(35,42)(36,46)(39,47)(40,48);(2,7)(6,14)(8,15)(9,16)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,33,34,37)(24,39,28,40)(25,41,32,42)(29,44,45,46)(38,47,43,48)

Polycyclic presentation:

2^9|a^2=g,d^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, 94465);