#8745866

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

4028 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)(3,9,10,6)(4,7,14,13)(11,16,12,15)(17,18)(19,28)(20,25)(21,24)(22,26)(23,27)(29,30)(31,32)(33,34)(35,42)(36,38)(37,41)(39,40)(43,44)(45,48)(46,47);(3,10)(6,9)(11,12)(15,16)(17,19,23,21)(18,24,27,28)(20,29,26,31)(22,30,25,32)(34,40)(36,45)(41,42)(43,46);(1,3,5,10)(2,6,8,9)(4,11,14,12)(7,15,13,16)(17,20,23,26)(18,25,27,22)(19,29,21,31)(24,32,28,30)(33,35,39,37)(34,41,40,42)(36,46,45,43)(38,44,48,47);(1,3)(2,6)(4,12)(5,10)(7,16)(8,9)(11,14)(13,15)(17,20)(18,25)(19,29)(21,31)(22,27)(23,26)(24,32)(28,30)(33,36,39,45)(34,38,40,48)(35,43,37,46)(41,47,42,44);(1,2)(3,6)(4,13)(5,8)(7,14)(9,10)(11,16)(12,15)(17,21,23,19)(18,24,27,28)(20,31,26,29)(22,30,25,32)(33,37,39,35)(34,42,40,41)(36,46,45,43)(38,44,48,47);(1,4,5,14)(2,7,8,13)(3,11,10,12)(6,15,9,16)(17,22,23,25)(18,26,27,20)(19,30,21,32)(24,31,28,29)(33,38)(34,36)(35,44)(37,47)(39,48)(40,45)(41,46)(42,43)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gi,d^2=hi,e^2=g,f^2=gh,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, 8653062);