#8549592

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

4090 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)(3,11)(4,13)(5,10)(6,7)(8,12)(9,14)(15,16)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(2,7)(3,12)(9,13)(10,16)(17,18,22,23)(19,24,28,31)(20,25,29,21)(26,27,32,30)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(2,7)(4,14)(8,11)(10,16)(18,23)(20,29)(24,31)(26,32)(33,35,39,36)(34,41,37,40)(38,43,46,44)(42,48,45,47);(1,3,6,12)(2,8,7,11)(4,10,14,16)(5,13,15,9)(17,19)(18,24)(20,26)(21,30)(22,28)(23,31)(25,27)(29,32)(33,36,39,35)(34,41,37,40)(38,43,46,44)(42,47,45,48);(1,4)(2,9)(3,10)(5,8)(6,14)(7,13)(11,15)(12,16)(17,20)(18,25)(19,26)(21,23)(22,29)(24,27)(28,32)(30,31)(33,37,39,34)(35,41,36,40)(38,45,46,42)(43,48,44,47);(1,5)(2,10)(3,9)(4,8)(6,15)(7,16)(11,14)(12,13)(17,21,22,25)(18,20,23,29)(19,27,28,30)(24,32,31,26)(33,38,39,46)(34,42,37,45)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=g,e^2=gh,f^2=ghi,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, 8456788);