#10147566

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4020 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:

(1,2,6,7)(3,11,12,8)(4,9,14,13)(5,15,16,10)(17,18)(19,21)(20,23)(22,28)(24,26)(25,30)(27,31)(29,32)(33,34)(35,41)(36,46)(37,39)(38,42)(40,43)(44,47)(45,48);(2,7)(4,14)(8,11)(10,15)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32)(33,35,40,37)(34,41,43,39)(36,45,38,44)(42,47,46,48);(2,7)(3,12)(9,13)(10,15)(18,20)(19,26)(25,28)(27,32)(33,36,40,38)(34,42,43,46)(35,44,37,45)(39,47,41,48);(1,3,6,12)(2,8,7,11)(4,15,14,10)(5,13,16,9)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,37,40,35)(34,39,43,41)(36,44,38,45)(42,48,46,47);(1,4,6,14)(2,9,7,13)(3,10,12,15)(5,8,16,11)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,38,40,36)(34,42,43,46)(35,45,37,44)(39,47,41,48);(1,5,6,16)(2,10,7,15)(3,13,12,9)(4,11,14,8)(17,22,23,30)(18,25,20,28)(19,27,26,32)(21,29,24,31)(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=h,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, 10054762);