#10147089

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	#51940	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4309 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=g,e^2=i,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, 10054285);