#8533190

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

3957 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gi,d^2=g,e^2=g,f^2=gh,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, 8440386);