#8696294

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

4176 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=i,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8603490);