#185903

order	512
nilpotency class	3
isoclinism	isoclinic class of #18346
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62788	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2157 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)(17,18,22,28,23,27,21,26)(19,29,31,42,32,24,37,39)(20,33,36,40,35,25,34,41)(30,43,44,48,45,38,46,47);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19)(18,24)(20,30)(21,31)(22,37)(23,32)(25,38)(26,39)(27,29)(28,42)(33,43)(34,44)(35,45)(36,46)(40,47)(41,48);(17,20,23,35)(18,25,27,33)(19,30,32,45)(21,36,22,34)(24,38,29,43)(26,41,28,40)(31,46,37,44)(39,48,42,47);(2,7)(6,14)(8,15)(9,16)(17,21,23,22)(18,26,27,28)(19,31,32,37)(20,34,35,36)(24,39,29,42)(25,40,33,41)(30,44,45,46)(38,47,43,48)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,e^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,d^c=dh,e^c=eh,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, 93099);