#9726292

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

4098 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)(3,5)(4,11)(6,14)(7,8)(9,16)(10,15)(12,13)(17,18,22,20)(19,25,27,23)(21,24,30,28)(26,32,29,31)(33,34,39,40)(35,36,37,41)(38,46,47,42)(43,48,45,44);(1,3,7,4)(2,5,8,11)(6,10,16,12)(9,13,14,15)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(34,40)(35,37)(42,46)(43,45);(2,8)(3,4)(9,14)(10,12)(18,20)(23,25)(24,28)(31,32)(33,35,39,37)(34,41,40,36)(38,43,47,45)(42,48,46,44);(1,4,7,3)(2,5,8,11)(6,12,16,10)(9,13,14,15)(17,20,22,18)(19,25,27,23)(21,28,30,24)(26,32,29,31)(33,36,39,41)(34,35,40,37)(38,44,47,48)(42,43,46,45);(1,5)(2,3)(4,8)(6,15)(7,11)(9,12)(10,14)(13,16)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,37,39,35)(34,41,40,36)(38,45,47,43)(42,48,46,44);(1,6,7,16)(2,9,8,14)(3,10,4,12)(5,13,11,15)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,38,39,47)(34,42,40,46)(35,43,37,45)(36,44,41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=h,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9633488);