#8097801

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×#548	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4224 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi

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, 8004997);