#8186833

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=gi,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, 8094029);