#8436674

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	#39430	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4425 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=h,f^2=g,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, 8343870);