#8734677

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

4096 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)(3,7)(4,13)(5,9)(6,10)(8,12)(11,16)(14,15)(17,18,23,24)(19,21,20,25)(22,26,30,31)(27,29,28,32)(33,34,39,43)(35,41,37,40)(36,42,47,38)(44,45,46,48);(1,3,6,11)(2,7,10,16)(4,5,12,15)(8,14,13,9)(18,24)(19,20)(26,31)(27,28)(33,35,39,37)(34,40,43,41)(36,44,47,46)(38,45,42,48);(3,11)(4,12)(7,16)(8,13)(17,19,23,20)(18,25,24,21)(22,27,30,28)(26,32,31,29)(33,35,39,37)(34,41,43,40)(36,46,47,44)(38,45,42,48);(1,4)(2,8)(3,5)(6,12)(7,14)(9,16)(10,13)(11,15)(17,20,23,19)(18,21,24,25)(22,28,30,27)(26,29,31,32)(33,36,39,47)(34,42,43,38)(35,44,37,46)(40,48,41,45);(1,2)(3,7)(4,8)(5,14)(6,10)(9,15)(11,16)(12,13)(17,21)(18,19)(20,24)(22,29)(23,25)(26,27)(28,31)(30,32)(33,37,39,35)(34,40,43,41)(36,46,47,44)(38,45,42,48);(1,5)(2,9)(3,12)(4,11)(6,15)(7,8)(10,14)(13,16)(17,22,23,30)(18,26,24,31)(19,27,20,28)(21,29,25,32)(33,38,39,42)(34,36,43,47)(35,45,37,48)(40,44,41,46)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8641873);