#10147574

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3853 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10054770);