#9062095

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8969291);