#8097149

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

4208 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004345);