#8434616

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) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8341812);