#8402355

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	#59650	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4476 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)(3,7)(4,8)(5,6)(9,10)(11,18)(12,13)(14,15)(16,21)(17,22)(19,20)(23,24)(26,31)(32,35)(33,38)(39,40);(2,6)(3,8)(10,15)(12,22)(16,18)(19,23)(25,26,30,31)(27,32,28,35)(29,33,37,38)(34,39,36,40);(1,3)(2,4)(5,8)(6,7)(10,15)(13,17)(16,18)(20,24)(25,27,30,28)(26,32,31,35)(29,36,37,34)(33,40,38,39);(1,3)(2,4)(5,8)(6,7)(9,11,14,21)(10,16,15,18)(12,23,22,19)(13,20,17,24)(25,26)(27,32)(28,35)(29,33)(30,31)(34,39)(36,40)(37,38);(1,4)(2,3)(5,7)(6,8)(9,12)(10,17)(11,19)(13,15)(14,22)(16,20)(18,24)(21,23)(25,28,30,27)(26,32,31,35)(29,36,37,34)(33,39,38,40);(9,13,14,17)(10,12,15,22)(11,20,21,24)(16,23,18,19)(25,29,30,37)(26,33,31,38)(27,34,28,36)(32,39,35,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8309551);