#9203719

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

4200 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)(17,18,22,21)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34)(35,41)(36,40)(37,39)(38,42)(43,48)(44,47)(45,46);(1,3)(2,7)(4,11)(5,12)(6,10)(8,9)(13,14)(15,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,32)(29,31)(33,35,38,37)(34,39,42,41)(36,44,46,43)(40,48,45,47);(3,10)(7,8)(11,13)(12,16)(20,28)(24,29)(26,32)(30,31)(33,34,38,42)(35,39,37,41)(36,40,46,45)(43,47,44,48);(1,3)(2,8)(4,13)(5,12)(6,10)(7,9)(11,14)(15,16)(17,20,22,28)(18,24,21,29)(19,26,27,32)(23,30,25,31)(33,34)(35,39)(36,45)(37,41)(38,42)(40,46)(43,48)(44,47);(1,4,6,14)(2,5,9,15)(3,11,10,13)(7,12,8,16)(17,21,22,18)(19,25,27,23)(20,24,28,29)(26,30,32,31)(33,36)(34,40)(35,43)(37,44)(38,46)(39,47)(41,48)(42,45);(1,5)(2,4)(3,12)(6,15)(7,11)(8,13)(9,14)(10,16)(17,19)(18,25)(20,26)(21,23)(22,27)(24,31)(28,32)(29,30)(33,37,38,35)(34,41,42,39)(36,43,46,44)(40,47,45,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=gh,e^2=ghi,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, 9110915);