#8097043

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

4468 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,e^2=h,f^2=h,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, 8004239);