#8228709

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

4202 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)(3,4)(5,9)(6,7)(8,11)(10,12)(13,14)(15,16)(17,18)(19,26)(20,30)(21,25)(22,23)(24,27)(28,29)(31,32)(33,34)(35,40)(36,42)(37,45)(38,44)(39,41)(43,48)(46,47);(2,7)(4,8)(9,14)(12,15)(18,23)(19,27)(20,25)(29,32)(33,35,39,38)(34,40,41,44)(36,43,47,45)(37,42,48,46);(5,13)(9,14)(10,16)(12,15)(18,23)(21,30)(24,26)(29,32)(34,41)(35,38)(42,46)(43,45);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,19)(18,24)(20,28)(21,29)(22,27)(23,26)(25,31)(30,32)(33,36,39,47)(34,42,41,46)(35,45,38,43)(37,44,48,40);(1,4,6,8)(2,3,7,11)(5,12,13,15)(9,10,14,16)(17,20,22,25)(18,21,23,30)(19,28,27,31)(24,29,26,32)(33,37,39,48)(34,43,41,45)(35,46,38,42)(36,44,47,40);(1,5,6,13)(2,9,7,14)(3,10,11,16)(4,12,8,15)(17,21)(18,25)(19,29)(20,23)(22,30)(24,31)(26,28)(27,32)(33,38,39,35)(34,44,41,40)(36,43,47,45)(37,42,48,46)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=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, 8135905);