#9203704

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4187 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9110900);