#9717627

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

4169 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:

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

Polycyclic presentation:

2^9|a^2=h,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, 9624823);