#10040460

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	#52649	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3949 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18,23,21)(19,26,22,24)(20,25,29,30)(27,32,28,31)(33,34)(35,44)(36,37)(38,48)(39,41)(40,45)(42,47)(43,46);(1,3,6,11)(2,7,8,16)(4,14,10,12)(5,13,9,15)(17,19,23,22)(18,24,21,26)(20,28,29,27)(25,32,30,31)(33,35,39,45)(34,40,41,44)(36,46,47,38)(37,48,42,43);(2,8)(3,11)(4,10)(13,15)(18,21)(20,29)(24,26)(27,28)(34,41)(35,45)(36,47)(43,48);(1,4,6,10)(2,9,8,5)(3,12,11,14)(7,13,16,15)(17,20,23,29)(18,25,21,30)(19,27,22,28)(24,31,26,32)(33,36)(34,42)(35,46)(37,41)(38,45)(39,47)(40,43)(44,48);(1,3)(2,7)(4,12)(5,15)(6,11)(8,16)(9,13)(10,14)(17,21,23,18)(19,24,22,26)(20,30,29,25)(27,31,28,32)(33,37,39,42)(34,36,41,47)(35,43,45,48)(38,44,46,40);(1,5,6,9)(2,10,8,4)(3,13,11,15)(7,12,16,14)(17,22,23,19)(18,26,21,24)(20,28,29,27)(25,32,30,31)(33,38,39,46)(34,43,41,48)(35,36,45,47)(37,44,42,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=gi,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, 9947656);