#8227371

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

3919 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)(19,28)(20,26)(21,22)(23,25)(24,27)(29,30)(31,32)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,41,44,48);(1,2,6,7)(3,5,9,14)(4,11,13,8)(10,16,15,12)(17,19,24,23)(18,25,27,28)(20,29,22,32)(21,30,26,31)(33,35,38,46)(34,39,42,43)(36,37,45,44)(40,41,47,48);(2,7)(5,14)(8,11)(12,16)(17,20,24,22)(18,26,27,21)(19,29,23,32)(25,31,28,30)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,48,43,41);(1,3,6,9)(2,5,7,14)(4,10,13,15)(8,12,11,16)(17,21)(18,20)(19,30)(22,27)(23,31)(24,26)(25,29)(28,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,48)(41,47)(42,43);(1,4,6,13)(2,8,7,11)(3,10,9,15)(5,12,14,16)(17,22,24,20)(18,21,27,26)(19,29,23,32)(25,31,28,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,48)(45,47);(1,5)(2,9)(3,7)(4,12)(6,14)(8,15)(10,11)(13,16)(17,23,24,19)(18,25,27,28)(20,29,22,32)(21,31,26,30)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,47,43,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=hi,e^2=gi,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, 8134567);