#7657998

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4228 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=i,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7565194);