#9230878

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	#43926	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4557 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=gh,e^2=hi,f^2=gh,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, 9138074);