#9229604

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	#47393	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4431 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=g,e^2=i,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, 9136800);