#8472411

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

4457 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)(3,5)(4,7)(6,8)(9,10,14,18)(11,13,17,15)(12,16,22,24)(19,21,20,23)(28,36)(32,37)(34,40)(38,39);(2,3)(6,7)(9,11,14,17)(10,13,18,15)(12,20,22,19)(16,23,24,21)(25,26)(27,31)(28,37)(29,33)(30,35)(32,36)(34,38)(39,40);(1,2,5,3)(4,6,8,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,40)(37,39);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,22)(23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,40)(37,38);(1,4,5,8)(2,6,3,7)(9,12,14,22)(10,16,18,24)(11,19,17,20)(13,21,15,23)(25,28)(26,32)(27,34)(29,36)(30,38)(31,39)(33,37)(35,40);(9,13)(10,17)(11,18)(12,21)(14,15)(16,20)(19,24)(22,23)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,37)(39,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=h,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, 8379607);