#8682234

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4455 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8589430);