#8750776

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

4095 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)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31)(33,34)(35,44)(36,42)(37,45)(38,40)(39,41)(43,47)(46,48);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(33,35,39,38)(34,40,41,44)(36,45,46,43)(37,42,47,48);(2,3)(7,9)(8,10)(15,16)(20,28)(25,30)(27,29)(31,32)(34,41)(36,46)(40,44)(43,45);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(24,31)(26,32)(33,36)(34,42)(35,43)(37,40)(38,45)(39,46)(41,48)(44,47);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,21,23,19)(18,24,22,26)(20,29,28,27)(25,31,30,32)(33,37)(34,43)(35,42)(36,40)(38,48)(39,47)(41,45)(44,46);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,22,23,18)(19,26,21,24)(20,25,28,30)(27,31,29,32)(33,38,39,35)(34,40,41,44)(36,43,46,45)(37,42,47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8657972);