#8437848

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	#38905	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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8345044);