#8429474

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

3889 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8336670);