#8097220

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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,e^2=gh,f^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, 8004416);