#8425347

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

4639 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332543);