#8256612

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

4054 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=g,d^2=gh,e^2=hi,f^2=ghi,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, 8163808);