#8456822

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4623 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)(3,8)(4,6)(5,7)(10,11)(12,19)(16,18)(20,24)(25,26,30,34)(27,35,36,31)(28,32,37,40)(29,33,39,38);(3,4)(6,8)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(25,27)(26,31)(28,33)(29,32)(30,36)(34,35)(37,38)(39,40);(1,3,5,4)(2,6,7,8)(25,26)(27,31)(28,32)(29,33)(30,34)(35,36)(37,40)(38,39);(1,4,5,3)(2,6,7,8)(9,11,14,10)(12,17,19,15)(13,16,21,18)(20,22,24,23)(25,26)(27,35)(28,32)(29,38)(30,34)(31,36)(33,39)(37,40);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28,30,37)(26,32,34,40)(27,33,36,38)(29,35,39,31);(1,2)(3,6)(4,8)(5,7)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29,30,39)(26,33,34,38)(27,32,36,40)(28,35,37,31)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=i,d^2=gh,e^2=hi,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, 8364018);