#8429414

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	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4377 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8336610);