#8633787

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	#54942	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4603 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)(16,19)(17,22)(20,21)(23,24)(25,26,29,32)(27,30,34,33)(28,31,37,36)(35,38,40,39);(2,6)(3,8)(10,15)(11,19)(12,22)(21,23)(25,26)(27,33)(28,31)(29,32)(30,34)(35,39)(36,37)(38,40);(1,3)(2,4)(5,8)(6,7)(10,15)(12,22)(16,18)(20,24)(27,34)(30,33)(35,40)(38,39);(1,3)(2,4)(5,8)(6,7)(9,11)(10,16)(12,20)(13,23)(14,19)(15,18)(17,21)(22,24)(25,26)(27,30)(28,36)(29,32)(31,37)(33,34)(35,39)(38,40);(1,4)(2,3)(5,7)(6,8)(9,12)(10,17)(11,20)(13,15)(14,22)(16,21)(18,23)(19,24)(25,27,29,34)(26,30,32,33)(28,35,37,40)(31,38,36,39);(9,13,14,17)(10,12,15,22)(11,21,19,23)(16,24,18,20)(25,28)(26,31)(27,35)(29,37)(30,38)(32,36)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8540983);