#9726390

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

4099 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)(3,11)(4,8)(5,9)(6,10)(7,14)(12,15)(13,16)(17,18)(19,27)(20,29)(21,26)(22,30)(23,25)(24,28)(31,32)(33,34)(35,41)(36,40)(37,39)(38,42)(43,47)(44,46)(45,48);(1,3)(2,7)(4,12)(5,13)(6,14)(8,9)(10,11)(15,16)(17,19,23,28)(18,24,25,27)(20,21,30,31)(22,26,29,32)(33,35,38,37)(34,39,42,41)(36,44,45,43)(40,47,48,46);(18,25)(19,28)(20,30)(26,32)(33,34,38,42)(35,39,37,41)(36,40,45,48)(43,46,44,47);(1,4,6,13)(2,8,10,16)(3,12,14,5)(7,9,11,15)(17,20)(18,22)(19,21)(23,30)(24,26)(25,29)(27,32)(28,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5,6,12)(2,9,10,15)(3,13,14,4)(7,8,11,16)(17,21)(18,26)(19,20)(22,24)(23,31)(25,32)(27,29)(28,30)(33,36,38,45)(34,40,42,48)(35,43,37,44)(39,46,41,47);(1,2)(3,7)(4,8)(5,15)(6,10)(9,12)(11,14)(13,16)(17,22)(18,20)(19,26)(21,24)(23,29)(25,30)(27,31)(28,32)(33,37,38,35)(34,41,42,39)(36,43,45,44)(40,46,48,47)

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=gh,e^2=gi,f^2=hi,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, 9633586);