#8533596

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

4091 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=hi,d^2=g,e^2=h,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, 8440792);