#8497872

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

4488 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=g,e^2=g,f^2=gi,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, 8405068);