#8534916

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

3941 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)(3,4)(5,10)(6,7)(8,9)(11,12)(13,16)(14,15)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(33,34,39,38)(35,42,37,40)(36,41,46,45)(43,48,44,47);(2,7)(3,9)(10,15)(11,13)(18,19)(23,25)(24,26)(30,31)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(2,7)(4,8)(10,15)(12,16)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,3,6,9)(2,8,7,4)(5,13,14,11)(10,12,15,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,37,39,35)(34,42,38,40)(36,44,46,43)(41,48,45,47);(1,4,6,8)(2,9,7,3)(5,12,14,16)(10,13,15,11)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,34,39,38)(35,40,37,42)(36,41,46,45)(43,47,44,48);(1,5,6,14)(2,10,7,15)(3,11,9,13)(4,12,8,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=i,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8442112);