#8192518

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

4142 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=gh,e^2=ghi,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, 8099714);