#9713767

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) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=h,e^2=gh,f^2=gh,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, 9620963);