#10038505

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	#44086	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3969 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9945701);