#8455724

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

4209 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)(9,10,15,18)(11,17,13,16)(12,21,23,14)(19,20,22,24)(26,30)(27,33)(32,37)(34,39);(2,3)(6,7)(9,11,15,13)(10,16,18,17)(12,19,23,22)(14,20,21,24)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,3,5,2)(4,7,8,6)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,4,5,8)(2,6,3,7)(9,12,15,23)(10,14,18,21)(11,19,13,22)(16,20,17,24)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(9,13,15,11)(10,17,18,16)(12,22,23,19)(14,24,21,20)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(9,14)(10,12)(11,20)(13,24)(15,21)(16,19)(17,22)(18,23)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=hi,d^2=gh,e^2=ghi,f^2=h,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, 8362920);