#8639896

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3957 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)(3,9)(4,6)(5,8)(7,10)(11,16)(12,14)(13,15)(17,18,22,25)(19,26,21,23)(20,24,29,32)(27,31,28,30)(33,34)(35,39)(36,46)(37,38)(40,41)(42,47)(43,44)(45,48);(2,4)(3,10)(8,12)(11,13)(17,19,22,21)(18,23,25,26)(20,28,29,27)(24,31,32,30)(33,35,40,45)(34,39,41,48)(36,43,37,42)(38,47,46,44);(2,4)(7,9)(8,12)(15,16)(20,29)(24,32)(27,28)(30,31)(33,36,40,37)(34,38,41,46)(35,42,45,43)(39,44,48,47);(1,3,6,10)(2,7,4,9)(5,13,14,11)(8,16,12,15)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,37,40,36)(34,38,41,46)(35,43,45,42)(39,44,48,47);(1,4,6,2)(3,9,10,7)(5,12,14,8)(11,16,13,15)(17,20,22,29)(18,24,25,32)(19,27,21,28)(23,30,26,31)(33,38)(34,36)(35,44)(37,41)(39,42)(40,46)(43,48)(45,47);(1,5,6,14)(2,8,4,12)(3,11,10,13)(7,15,9,16)(17,21,22,19)(18,23,25,26)(20,28,29,27)(24,30,32,31)(33,39,40,48)(34,35,41,45)(36,44,37,47)(38,42,46,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8547092);