#9039445

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

4166 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=gi,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, 8946641);