#8188192

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

4008 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)(3,7)(4,9)(5,13)(6,8)(10,14)(11,12)(15,16)(18,23)(19,26)(20,28)(29,31)(33,34)(35,46)(36,39)(37,44)(38,43)(40,45)(41,48)(42,47);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,15,13,12)(17,18,22,23)(19,21,26,24)(20,25,28,30)(27,29,32,31)(33,35)(34,41)(36,37)(38,47)(39,43)(40,48)(42,44)(45,46);(2,8)(3,9)(10,13)(11,16)(17,19,22,26)(18,24,23,21)(20,27,28,32)(25,31,30,29)(33,36,40,47)(34,42,45,39)(35,37,48,38)(41,44,46,43);(1,4,6,7)(2,9,8,3)(5,12,14,15)(10,16,13,11)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,37,40,38)(34,43,45,44)(35,36,48,47)(39,46,42,41);(1,5,6,14)(2,10,8,13)(3,11,9,16)(4,12,7,15)(17,20,22,28)(18,25,23,30)(19,27,26,32)(21,29,24,31)(33,38,40,37)(34,44,45,43)(35,47,48,36)(39,41,42,46);(1,2)(3,7)(4,9)(5,10)(6,8)(11,15)(12,16)(13,14)(17,21,22,24)(18,26,23,19)(20,29,28,31)(25,32,30,27)(33,39,40,42)(34,36,45,47)(35,43,48,44)(37,46,38,41)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=gi,e^2=h,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, 8095388);