#9243825

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	#52365	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3842 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)(3,11)(4,14)(5,8)(6,10)(7,9)(12,15)(13,16)(17,18,24,28)(19,21,22,25)(20,27,23,26)(29,30,32,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,3,8,7)(2,9,5,11)(4,13,6,12)(10,16,14,15)(17,19)(18,25)(20,29)(21,28)(22,24)(23,32)(26,30)(27,31)(34,38)(39,41)(40,45)(46,47);(1,4,8,6)(2,10,5,14)(3,12,7,13)(9,15,11,16)(17,20,24,23)(18,26,28,27)(19,29,22,32)(21,31,25,30)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,5,8,2)(3,11,7,9)(4,10,6,14)(12,15,13,16)(17,21,24,25)(18,19,28,22)(20,30,23,31)(26,32,27,29)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,6,8,4)(2,10,5,14)(3,13,7,12)(9,15,11,16)(17,22,24,19)(18,25,28,21)(20,29,23,32)(26,31,27,30);(1,7,8,3)(2,11,5,9)(4,13,6,12)(10,16,14,15)(17,23,24,20)(18,27,28,26)(19,29,22,32)(21,31,25,30)(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=ghi,c^2=h,d^2=g,e^2=h,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, 9151021);