#8463795

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

4020 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31)(35,41)(38,42)(43,47)(46,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(20,28)(25,30)(27,29)(31,32)(33,34,37,40)(35,42,41,38)(36,39,45,44)(43,48,47,46);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(33,35,37,41)(34,38,40,42)(36,43,45,47)(39,46,44,48);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(24,31)(26,32)(33,34)(35,38)(36,44)(37,40)(39,45)(41,42)(43,48)(46,47);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,21,23,19)(18,26,22,24)(20,29,28,27)(25,32,30,31);(1,2)(3,7)(4,8)(5,14)(6,9)(10,13)(11,15)(12,16)(17,22,23,18)(19,26,21,24)(20,30,28,25)(27,32,29,31)(33,36)(34,39)(35,43)(37,45)(38,46)(40,44)(41,47)(42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gh,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8370991);