#9243497

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	#51768	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3861 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9150693);