#9039143

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
prime radical	#37245	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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)(3,8)(4,12)(5,10)(6,7)(9,13)(11,16)(14,15)(17,18)(19,26)(20,30)(21,25)(22,29)(23,28)(24,31)(27,32)(33,34,39,40)(35,43,44,41)(36,47,48,37)(38,45,46,42);(1,3,7,5)(2,8,6,10)(4,11,13,14)(9,15,12,16)(17,19)(18,24)(20,27)(21,29)(22,30)(23,31)(25,32)(26,28)(34,40)(35,44)(37,47)(42,45);(2,6)(3,5)(9,12)(11,14)(17,20,23,21)(18,25,28,30)(19,27,31,29)(22,24,32,26)(33,35,39,44)(34,41,40,43)(36,45,48,42)(37,46,47,38);(1,4,7,13)(2,9,6,12)(3,11,5,14)(8,15,10,16)(17,21,23,20)(18,25,28,30)(19,29,31,27)(22,24,32,26)(33,36)(34,37)(35,45)(38,43)(39,48)(40,47)(41,46)(42,44);(1,5,7,3)(2,10,6,8)(4,11,13,14)(9,15,12,16)(17,19)(18,26)(20,29)(21,27)(22,30)(23,31)(24,28)(25,32)(33,37)(34,36)(35,46)(38,44)(39,47)(40,48)(41,45)(42,43);(1,6,7,2)(3,10,5,8)(4,12,13,9)(11,16,14,15)(17,22,23,32)(18,27,28,29)(19,30,31,25)(20,26,21,24)(33,38,39,46)(34,42,40,45)(35,37,44,47)(36,43,48,41)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,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, 8946339);