#9224354

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	#49814	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4008 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)(3,8)(4,14)(5,6)(7,10)(9,13)(11,16)(12,15)(17,18,24,27)(19,28,32,25)(20,22,21,31)(23,26,30,29)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(1,3)(2,8)(4,15)(5,11)(6,16)(7,13)(9,10)(12,14)(17,19)(18,25)(20,29)(21,26)(22,30)(23,31)(24,32)(27,28)(36,44)(39,46)(42,48)(45,47);(1,4,7,5)(2,6,10,14)(3,11,13,15)(8,12,9,16)(17,20,24,21)(18,22,27,31)(19,29,32,26)(23,25,30,28)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,5,7,4)(2,6,10,14)(3,11,13,15)(8,16,9,12)(17,21,24,20)(18,22,27,31)(19,26,32,29)(23,25,30,28);(1,6)(2,5)(3,12)(4,10)(7,14)(8,15)(9,11)(13,16)(17,22)(18,21)(19,30)(20,27)(23,32)(24,31)(25,26)(28,29)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48);(1,3)(2,9)(4,11)(5,15)(6,16)(7,13)(8,10)(12,14)(17,23,24,30)(18,26,27,29)(19,31,32,22)(20,25,21,28)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9131550);