#9713766

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

4106 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)(3,6)(4,7)(5,8)(9,10)(11,16)(12,15)(13,14)(17,18,23,20)(19,27,28,24)(21,25,32,29)(22,30,31,26)(33,34)(35,41)(36,37)(38,42)(39,48)(40,44)(43,45)(46,47);(1,3)(2,6)(4,10)(5,13)(7,9)(8,14)(11,15)(12,16)(17,19,23,28)(18,24,20,27)(21,31,32,22)(25,30,29,26)(33,35)(34,41)(36,43)(37,45)(38,48)(39,42)(40,46)(44,47);(2,4)(3,9)(8,12)(11,13)(18,20)(24,27)(25,29)(26,30)(33,36,40,38)(34,42,44,37)(35,43,46,48)(39,47,45,41);(1,4,7,2)(3,10,9,6)(5,12,15,8)(11,14,13,16)(17,20,23,18)(19,27,28,24)(21,29,32,25)(22,30,31,26)(33,37,40,42)(34,36,44,38)(35,45,46,39)(41,43,47,48);(1,5,7,15)(2,8,4,12)(3,11,9,13)(6,16,10,14)(17,21,23,32)(18,25,20,29)(19,22,28,31)(24,26,27,30)(33,38,40,36)(34,42,44,37)(35,43,46,48)(39,41,45,47);(1,6,7,10)(2,9,4,3)(5,14,15,16)(8,11,12,13)(17,22)(18,26)(19,21)(20,30)(23,31)(24,25)(27,29)(28,32)(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=h,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, 9620962);