#10040608

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	#54967	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,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, 9947804);