#9711984

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

4131 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)(3,9,10,4)(5,13,15,8)(11,12,16,14)(17,18)(19,24)(20,22)(21,26)(23,25)(27,32)(28,29)(30,31)(34,38)(36,44)(39,41)(42,48);(2,7)(4,9)(8,13)(12,14)(17,19)(18,24)(20,28)(21,25)(22,29)(23,26)(27,31)(30,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(2,7)(3,10)(8,13)(11,16)(18,25)(19,26)(20,27)(29,30)(33,34,37,38)(35,39,43,41)(36,40,44,45)(42,46,48,47);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,14)(15,16)(17,20,23,27)(18,22,25,32)(19,28,26,31)(21,30,24,29)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,4)(2,3)(5,14)(6,9)(7,10)(8,16)(11,13)(12,15)(17,21,23,24)(18,26,25,19)(20,30,27,29)(22,31,32,28)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5,6,15)(2,8,7,13)(3,11,10,16)(4,12,9,14)(17,22)(18,27)(19,29)(20,25)(21,28)(23,32)(24,31)(26,30)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,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, 9619180);