#8456821

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

4542 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)(10,14)(11,17)(16,22)(18,19)(25,26,30,32)(27,33,35,29)(28,31,37,38)(34,39,40,36);(3,4)(6,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,27,30,35)(26,29,32,33)(28,34,37,40)(31,36,38,39);(1,3,5,4)(2,6,7,8)(25,26)(27,29)(28,31)(30,32)(33,35)(34,36)(37,38)(39,40);(1,4,5,3)(2,6,7,8)(9,11,13,17)(10,15,14,21)(12,19,20,18)(16,24,22,23)(25,26)(27,33)(28,31)(29,35)(30,32)(34,39)(36,40)(37,38);(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,28,30,37)(26,31,32,38)(27,34,35,40)(29,36,33,39);(1,2)(3,6)(4,8)(5,7)(9,12,13,20)(10,16,14,22)(11,18,17,19)(15,23,21,24)(25,29,30,33)(26,27,32,35)(28,36,37,39)(31,34,38,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,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, 8364017);