#9333322

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	#49637	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3936 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=ghi,e^2=h,f^2=gi,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, 9240518);