#8493048

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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8400244);