#8227339

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3804 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:

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

Polycyclic presentation:

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

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, 8134535);