#8097013

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

4628 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004209);