#9221176

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	#49754	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=gh,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, 9128372);