#10147572

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

3842 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:

(1,2,6,8)(3,10,5,7)(4,9,14,13)(11,16,12,15)(17,18)(19,29)(20,26)(21,22)(23,27)(24,28)(25,31)(30,32)(33,34)(35,41)(36,46)(37,39)(38,42)(40,43)(44,47)(45,48);(1,3,6,5)(2,7,8,10)(4,12,14,11)(9,16,13,15)(17,19)(18,25)(20,23)(21,30)(22,27)(24,31)(26,32)(28,29)(33,35,40,37)(34,41,43,39)(36,45,38,44)(42,47,46,48);(2,8)(7,10)(9,13)(15,16)(17,20,24,21)(18,26,28,22)(19,23,31,30)(25,32,29,27)(33,36,40,38)(34,42,43,46)(35,44,37,45)(39,47,41,48);(1,2)(3,7)(4,13)(5,10)(6,8)(9,14)(11,16)(12,15)(17,21,24,20)(18,26,28,22)(19,30,31,23)(25,32,29,27)(33,37,40,35)(34,39,43,41)(36,44,38,45)(42,48,46,47);(1,4,6,14)(2,9,8,13)(3,11,5,12)(7,15,10,16)(17,22,24,26)(18,21,28,20)(19,27,31,32)(23,25,30,29)(33,38,40,36)(34,42,43,46)(35,45,37,44)(39,47,41,48);(1,5,6,3)(2,10,8,7)(4,11,14,12)(9,15,13,16)(17,23,24,30)(18,27,28,32)(19,20,31,21)(22,29,26,25)(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=h,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, 10054768);