#8227361

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

3782 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)(19,25,20,22)(21,29,31,26)(27,30,28,32)(33,34)(35,42)(36,37)(38,40)(39,43)(41,48)(44,45)(46,47);(1,2,6,7)(3,5,9,14)(4,11,13,8)(10,16,15,12)(18,24)(19,20)(26,29)(27,28)(33,35,39,38)(34,40,43,42)(36,46,41,44)(37,45,48,47);(2,7)(5,14)(8,11)(12,16)(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,36,39,41)(34,37,43,48)(35,44,38,46)(40,47,42,45);(1,3,6,9)(2,5,7,14)(4,10,13,15)(8,12,11,16)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,37)(34,41)(35,45)(36,43)(38,47)(39,48)(40,44)(42,46);(1,4,6,13)(2,8,7,11)(3,10,9,15)(5,12,14,16)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,35,39,38)(34,42,43,40)(36,44,41,46)(37,45,48,47);(1,5)(2,9)(3,7)(4,12)(6,14)(8,15)(10,11)(13,16)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,38,39,35)(34,40,43,42)(36,46,41,44)(37,47,48,45)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,d^2=ghi,e^2=i,f^2=g,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, 8134557);