#9726092

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

4108 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)(3,5)(4,11)(6,14)(7,8)(9,16)(10,15)(12,13)(17,18)(19,25)(20,21)(22,24)(23,29)(26,31)(27,30)(28,32)(34,36)(35,38)(41,44)(43,45);(1,3,7,4)(2,5,8,11)(6,10,16,12)(9,13,14,15)(17,19,22,29)(18,23,24,25)(20,27,26,32)(21,28,31,30)(33,34,39,36)(35,42,38,40)(37,44,46,41)(43,47,45,48);(2,8)(3,4)(9,14)(10,12)(18,24)(20,26)(23,25)(27,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,4,7,3)(2,5,8,11)(6,12,16,10)(9,13,14,15)(17,20,22,26)(18,21,24,31)(19,27,29,32)(23,28,25,30)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,5)(2,3)(4,8)(6,15)(7,11)(9,12)(10,14)(13,16)(17,19)(18,25)(20,27)(21,30)(22,29)(23,24)(26,32)(28,31)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,6,7,16)(2,9,8,14)(3,10,4,12)(5,13,11,15)(17,21)(18,26)(19,28)(20,24)(22,31)(23,32)(25,27)(29,30)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=h,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, 9633288);