#8821931

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

4079 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8729127);