#7651303

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	#39853	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4482 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7558499);