#8167376

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) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8074572);