#10147797

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

4153 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,e^2=ghi,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, 10054993);