#9039425

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

4143 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,d^2=i,e^2=gh,f^2=g,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, 8946621);