#9716863

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

4112 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)(3,5)(4,13)(6,15)(7,8)(9,14)(10,12)(11,16)(17,18)(19,28)(20,23)(21,26)(22,25)(24,27)(29,32)(30,31)(36,44)(40,47)(42,48)(45,46);(1,3,7,11)(2,5,8,16)(4,12,14,6)(9,15,13,10)(17,19,24,22)(18,25,27,28)(20,29,21,30)(23,31,26,32)(33,34)(35,37)(36,40)(38,41)(39,43)(42,45)(44,47)(46,48);(2,8)(3,11)(4,14)(10,15)(17,20,24,21)(18,23,27,26)(19,29,22,30)(25,31,28,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,48)(45,47);(1,4)(2,9)(3,12)(5,15)(6,11)(7,14)(8,13)(10,16)(17,21,24,20)(18,26,27,23)(19,30,22,29)(25,32,28,31)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,5)(2,3)(4,15)(6,13)(7,16)(8,11)(9,12)(10,14)(17,22,24,19)(18,25,27,28)(20,30,21,29)(23,31,26,32)(33,37,38,39)(34,35,41,43)(36,45,44,46)(40,42,47,48);(1,6,7,12)(2,10,8,15)(3,4,11,14)(5,9,16,13)(17,23,24,26)(18,20,27,21)(19,31,22,32)(25,29,28,30)(33,35)(34,37)(36,42)(38,43)(39,41)(40,45)(44,48)(46,47)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=g,e^2=ghi,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, 9624059);