#9127539

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4025 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9034735);