#10168462

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

4083 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=i,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10075658);