#8873795

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

3960 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=hi,d^2=gi,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8780991);