#8114382

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

4119 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8021578);