#8464402

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

4549 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,e^2=hi,f^2=gi,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, 8371598);