#8463215

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4154 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=h,d^2=gh,e^2=ghi,f^2=gi,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, 8370411);