#8463710

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	#34889	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4346 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)(3,6)(4,7)(5,8)(9,10,15,18)(11,19,22,16)(12,23,21,14)(13,24,20,17)(25,26,31,30)(27,32,37,36)(28,38,33,29)(34,40,39,35);(3,8)(5,6)(9,11)(10,16)(12,20)(13,21)(14,17)(15,22)(18,19)(23,24)(25,27,31,37)(26,32,30,36)(28,34,33,39)(29,35,38,40);(1,3)(2,5)(4,8)(6,7)(25,28)(26,29)(27,34)(30,38)(31,33)(32,35)(36,40)(37,39);(1,3)(2,6)(4,8)(5,7)(9,12,15,21)(10,14,18,23)(11,20,22,13)(16,17,19,24)(25,27)(26,32)(28,34)(29,35)(30,36)(31,37)(33,39)(38,40);(9,13,15,20)(10,17,18,24)(11,21,22,12)(14,19,23,16)(25,29,31,38)(26,33,30,28)(27,35,37,40)(32,39,36,34);(1,2)(3,5)(4,7)(6,8)(9,14)(10,12)(11,17)(13,19)(15,23)(16,20)(18,21)(22,24)(25,30,31,26)(27,36,37,32)(28,38,33,29)(34,40,39,35)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,d^2=g,e^2=ghi,f^2=hi,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, 8370906);