#8188522

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

3975 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:

(1,2,6,10)(3,8,4,7)(5,9,13,16)(11,15,12,14)(18,23)(24,25)(26,30)(31,32)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,41,44,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,18,22,23)(19,24,20,25)(21,26,29,30)(27,31,28,32)(33,35,38,46)(34,39,42,43)(36,37,45,44)(40,41,47,48);(1,3,6,4)(2,7,10,8)(5,12,13,11)(9,15,16,14)(17,19,22,20)(18,24,23,25)(21,28,29,27)(26,32,30,31)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,48,43,41);(1,4,6,3)(2,8,10,7)(5,12,13,11)(9,15,16,14)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,35)(34,39)(36,44)(37,45)(38,46)(40,48)(41,47)(42,43);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,20,22,19)(18,25,23,24)(21,28,29,27)(26,32,30,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,48)(45,47);(1,5,6,13)(2,9,10,16)(3,11,4,12)(7,14,8,15)(17,21,22,29)(18,26,23,30)(19,27,20,28)(24,31,25,32)(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=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095718);