#9039386

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

4139 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)(17,18,23,21)(19,24,20,25)(22,26,30,29)(27,31,28,32)(35,36)(38,42)(43,44)(45,47);(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(18,21)(24,25)(26,29)(31,32)(33,34,39,41)(35,42,36,38)(37,40,46,48)(43,47,44,45);(2,6)(4,12)(7,10)(11,15)(17,19,23,20)(18,24,21,25)(22,28,30,27)(26,32,29,31)(33,35,39,36)(34,38,41,42)(37,44,46,43)(40,47,48,45);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,36,39,35)(34,38,41,42)(37,43,46,44)(40,47,48,45);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,21,23,18)(19,24,20,25)(22,29,30,26)(27,31,28,32)(33,37,39,46)(34,40,41,48)(35,43,36,44)(38,45,42,47);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)(33,38)(34,35)(36,41)(37,45)(39,42)(40,43)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,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, 8946582);