#8718038

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

4111 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8625234);