#8258124

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

4069 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)(3,8,10,4)(5,13,15,9)(11,12,14,16)(17,18,24,28)(19,29,31,25)(20,22,21,26)(23,27,30,32)(33,34,38,37)(35,39,41,45)(36,40,43,44)(42,46,47,48);(1,3)(2,8)(4,6)(5,14)(7,10)(9,12)(11,15)(13,16)(17,19)(18,25)(20,27)(21,32)(22,23)(24,31)(26,30)(28,29)(35,41)(36,43)(39,45)(40,44);(2,6)(3,10)(9,13)(11,14)(17,20,24,21)(18,26,28,22)(19,27,31,32)(23,25,30,29)(33,35,38,41)(34,39,37,45)(36,42,43,47)(40,46,44,48);(1,4)(2,3)(5,12)(6,10)(7,8)(9,11)(13,14)(15,16)(17,21,24,20)(18,22,28,26)(19,27,31,32)(23,25,30,29);(1,5,7,15)(2,9,6,13)(3,11,10,14)(4,12,8,16)(17,22)(18,20)(19,30)(21,28)(23,31)(24,26)(25,32)(27,29)(33,36,38,43)(34,40,37,44)(35,42,41,47)(39,46,45,48);(1,6,7,2)(3,4,10,8)(5,13,15,9)(11,12,14,16)(17,23,24,30)(18,27,28,32)(19,22,31,26)(20,29,21,25)(33,37,38,34)(35,39,41,45)(36,44,43,40)(42,46,47,48)

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=g,e^2=i,f^2=gi,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, 8165320);