#9203757

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	#50527	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4085 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9110953);