#9062149

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	#44158	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4161 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=g,d^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, 8969345);