#10407788

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

4272 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,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, 10314984);