#10068902

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

4313 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)(19,22)(20,23)(21,26)(24,30)(25,27)(28,32)(29,31)(33,34)(35,38)(36,39)(37,42)(40,45)(41,44)(43,47)(46,48);(2,6)(3,9)(8,11)(10,12)(17,18)(19,22)(20,25)(21,24)(23,27)(26,30)(28,32)(29,31)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(2,6)(3,9)(8,11)(10,12)(18,19)(20,27)(24,26)(28,29)(34,35)(39,41)(40,42)(46,47);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,19,22,18)(20,25,27,23)(21,26,30,24)(28,32,29,31)(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)(17,20,22,27)(18,23,19,25)(21,29,30,28)(24,32,26,31)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,4,5,13)(2,8,6,11)(3,10,9,12)(7,15,14,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)

Polycyclic presentation:

2^9|b^2=ghi,d^2=i,e^2=gi,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, 9976098);