#8581812

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

4081 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,d^2=i,e^2=ghi,f^2=hi,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, 8489008);