#10023076

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

4105 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)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,18)(19,25)(20,24)(21,26)(22,23)(27,32)(28,29)(30,31)(33,34)(35,39)(36,37)(38,40)(41,42)(43,48)(44,47)(45,46);(2,6)(3,9)(8,11)(10,12)(17,19)(18,21)(20,28)(22,26)(23,25)(24,32)(27,30)(29,31)(33,35,38,43)(34,39,40,48)(36,46,42,44)(37,45,41,47);(2,6)(3,9)(8,11)(10,12)(18,23)(19,26)(24,31)(27,28)(34,40)(35,43)(36,42)(45,47);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,19)(18,21)(20,27)(22,26)(23,25)(24,29)(28,30)(31,32)(33,36,38,42)(34,41,40,37)(35,44,43,46)(39,45,48,47);(1,3,5,9)(2,7,6,14)(4,12,13,10)(8,16,11,15)(17,20,22,30)(18,24,23,31)(19,27,26,28)(21,29,25,32)(33,35)(34,39)(36,44)(37,47)(38,43)(40,48)(41,45)(42,46);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,21)(18,19)(20,29)(22,25)(23,26)(24,27)(28,31)(30,32)(33,37,38,41)(34,42,40,36)(35,45,43,47)(39,44,48,46)

Polycyclic presentation:

2^9|b^2=gh,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9930272);