#9039132

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

3971 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)(17,18,22,20)(19,25,27,23)(21,28,29,24)(26,30,32,31)(33,34)(35,40)(36,37)(38,48)(39,42)(41,46)(43,47)(44,45);(2,7)(4,9)(8,15)(10,13)(17,18,22,20)(19,23,27,25)(21,24,29,28)(26,30,32,31)(33,35,39,38)(34,40,42,48)(36,43,46,45)(37,47,41,44);(4,9)(5,14)(10,13)(11,16)(18,20)(21,29)(23,25)(26,32)(33,36)(34,41)(35,43)(37,42)(38,45)(39,46)(40,44)(47,48);(1,3,6,12)(2,8,7,15)(4,13,9,10)(5,11,14,16)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31);(1,4,6,9)(2,5,7,14)(3,10,12,13)(8,16,15,11)(17,20,22,18)(19,23,27,25)(21,28,29,24)(26,30,32,31)(33,37,39,41)(34,36,42,46)(35,44,38,47)(40,45,48,43);(1,5)(2,9)(3,11)(4,7)(6,14)(8,10)(12,16)(13,15)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,38,39,35)(34,40,42,48)(36,45,46,43)(37,47,41,44)

Polycyclic presentation:

2^9|a^2=gi,b^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, 8946328);