#10040986

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4297 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)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,34)(35,38)(36,39)(37,42)(40,45)(41,44)(43,47)(46,48);(2,6)(3,9)(8,11)(10,12)(20,27)(24,30)(26,31)(28,32)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(2,6)(3,9)(8,11)(10,12)(18,23)(19,25)(24,30)(26,31)(34,35)(39,41)(40,42)(46,47);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,3,5,9)(2,7,6,14)(4,12,13,10)(8,16,11,15)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,4,5,13)(2,8,6,11)(3,10,9,12)(7,15,14,16)(17,21)(18,19)(20,28)(22,29)(23,25)(24,26)(27,32)(30,31)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=i,e^2=g,f^2=g,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, 9948182);