#10247628

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

4004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154824);