#8438054

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

4865 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=g,e^2=gi,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, 8345250);