#9713453

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

4100 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)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18)(19,22)(20,21)(23,31)(24,26)(25,32)(27,28)(29,30)(34,39)(35,42)(37,46)(44,47);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,24,21)(18,20,26,22)(23,27,32,29)(25,28,31,30)(34,39)(36,41)(40,43)(44,47);(2,3)(7,9)(8,10)(14,15)(17,20,24,22)(18,19,26,21)(23,28,32,30)(25,27,31,29)(33,34,38,39)(35,43,42,40)(36,37,41,46)(44,48,47,45);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,21,24,19)(18,20,26,22)(23,29,32,27)(25,28,31,30)(33,35,38,42)(34,40,39,43)(36,44,41,47)(37,45,46,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,22,24,20)(18,19,26,21)(23,28,32,30)(25,29,31,27)(33,36)(34,37)(35,44)(38,41)(39,46)(40,45)(42,47)(43,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,23)(18,25)(19,27)(20,28)(21,29)(22,30)(24,32)(26,31)(33,37)(34,41)(35,45)(36,39)(38,46)(40,47)(42,48)(43,44)

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=h,e^2=g,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, 9620649);