#9625418

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

4530 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9532614);