#8097015

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

4581 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=g,e^2=gh,f^2=g,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, 8004211);