#8496548

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

4553 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^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, 8403744);