#9163980

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3870 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=g,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9071176);