#10246958

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

4070 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=i,e^2=h,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, 10154154);