#8227338

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,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, 8134534);