#9039158

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

4048 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=gi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8946354);