#8492784

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

4441 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

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, 8399980);