#10407787

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

4124 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)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18,23,20)(19,24,26,21)(22,25,31,28)(27,32,30,29)(33,34)(35,36)(37,45)(38,40)(39,47)(41,43)(42,48)(44,46);(2,3)(5,13)(7,9)(12,16)(17,19)(18,24)(20,21)(22,30)(23,26)(25,29)(27,31)(28,32)(33,35,38,43)(34,36,40,41)(37,46,47,42)(39,48,45,44);(4,11)(7,9)(12,16)(14,15)(19,26)(21,24)(27,30)(29,32)(33,34,38,40)(35,41,43,36)(37,39,47,45)(42,48,46,44);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,20,23,18)(19,24,26,21)(22,28,31,25)(27,32,30,29)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,46)(47,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,21,23,24)(18,19,20,26)(22,29,31,32)(25,27,28,30)(33,34)(35,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22,23,31)(18,25,20,28)(19,27,26,30)(21,29,24,32)(33,37)(34,39)(35,42)(36,44)(38,47)(40,45)(41,48)(43,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=ghi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10314983);