#9716227

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

3964 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=i,f^2=ghi,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, 9623423);