#10040813

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	#2733	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3867 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)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,18,23,25)(19,29,30,24)(20,21,27,26)(22,32,31,28)(33,34,38,39)(35,40,45,48)(36,41,46,44)(37,42,47,43);(2,6)(3,9)(8,11)(10,12)(17,19)(18,24)(20,31)(21,32)(22,27)(23,30)(25,29)(26,28)(34,39)(36,46)(40,48)(42,43);(2,6)(3,9)(8,11)(10,12)(18,25)(19,30)(20,27)(28,32)(33,35,38,45)(34,40,39,48)(36,43,46,42)(37,44,47,41);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,20,23,27)(18,26,25,21)(19,31,30,22)(24,28,29,32)(33,36)(34,41)(35,43)(37,40)(38,46)(39,44)(42,45)(47,48);(1,3,5,9)(2,7,6,14)(4,12,13,10)(8,16,11,15)(17,21,23,26)(18,27,25,20)(19,28,30,32)(22,24,31,29)(33,35)(34,40)(36,43)(37,41)(38,45)(39,48)(42,46)(44,47);(1,4,5,13)(2,8,6,11)(3,10,9,12)(7,15,14,16)(17,22,23,31)(18,28,25,32)(19,27,30,20)(21,24,26,29)(33,37)(34,42)(35,44)(36,40)(38,47)(39,43)(41,45)(46,48)

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,d^2=i,e^2=h,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, 9948009);