#10247455

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

4320 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)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18)(19,20)(21,29)(22,23)(24,31)(25,26)(27,30)(28,32)(34,38)(36,44)(39,41)(42,45);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,18)(19,25)(20,26)(21,24)(22,23)(27,30)(28,32)(29,31)(33,34,37,38)(35,41,43,39)(36,40,44,46)(42,48,45,47);(2,7)(6,13)(8,11)(14,15)(18,23)(19,26)(24,29)(27,32)(33,34,37,38)(35,39,43,41)(36,40,44,46)(42,47,45,48);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,19,22,26)(18,20,23,25)(21,27,31,32)(24,28,29,30)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,46)(45,47);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,20)(18,19)(21,30)(22,25)(23,26)(24,32)(27,29)(28,31)(33,35,37,43)(34,39,38,41)(36,45,44,42)(40,48,46,47);(1,4,5,12)(2,8,7,11)(3,9,10,16)(6,14,13,15)(17,21,22,31)(18,24,23,29)(19,27,26,32)(20,28,25,30)(33,36)(34,40)(35,42)(37,44)(38,46)(39,47)(41,48)(43,45)

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=i,e^2=g,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, 10154651);