#8434728

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

4665 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)(3,7)(4,5)(6,8)(10,11)(12,19)(16,18)(20,24)(25,26,30,34)(27,32,35,31)(28,29,33,39)(36,38,40,37);(2,5)(6,7)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(27,35)(31,32)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(25,27)(26,31)(28,36)(29,37)(30,35)(32,34)(33,40)(38,39);(1,3,4,8)(2,6,5,7)(9,11,14,10)(12,17,19,15)(13,16,21,18)(20,22,24,23)(25,27)(26,32)(28,36)(29,38)(30,35)(31,34)(33,40)(37,39);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28,30,33)(26,29,34,39)(27,36,35,40)(31,37,32,38);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,33)(27,37)(28,34)(30,39)(31,40)(32,36)(35,38)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=g,e^2=h,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, 8341924);