#8617778

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

4108 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)(3,9)(4,6)(5,8)(7,10)(11,16)(12,14)(13,15)(17,18,23,27)(19,24,21,25)(20,22,26,32)(28,29,30,31)(33,34,40,43)(35,44,46,41)(36,39,37,47)(38,42,48,45);(2,4)(3,10)(8,12)(11,13)(17,19)(18,24)(20,28)(21,23)(22,29)(25,27)(26,30)(31,32)(33,35,40,46)(34,41,43,44)(36,42,37,45)(38,47,48,39);(2,4)(7,9)(8,12)(15,16)(19,21)(24,25)(28,30)(29,31)(33,36,40,37)(34,39,43,47)(35,42,46,45)(38,44,48,41);(1,3)(2,7)(4,9)(5,13)(6,10)(8,16)(11,14)(12,15)(17,20)(18,22)(19,28)(21,30)(23,26)(24,29)(25,31)(27,32)(33,37,40,36)(34,39,43,47)(35,45,46,42)(38,44,48,41);(1,4,6,2)(3,9,10,7)(5,12,14,8)(11,16,13,15)(17,21,23,19)(18,25,27,24)(20,30,26,28)(22,31,32,29)(33,38)(34,42)(35,39)(36,41)(37,44)(40,48)(43,45)(46,47);(1,5,6,14)(2,8,4,12)(3,11,10,13)(7,15,9,16)(17,22)(18,26)(19,29)(20,27)(21,31)(23,32)(24,30)(25,28)(33,39)(34,37)(35,38)(36,43)(40,47)(41,45)(42,44)(46,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8524974);