#8097045

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

4488 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)(12,17)(13,18)(14,15)(16,20)(21,23)(22,24)(25,26,30,34)(27,31,28,32)(29,33,37,40)(35,38,36,39);(2,5)(6,7)(10,15)(11,20)(17,24)(18,21)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,21)(13,17)(14,20)(15,19)(18,22)(23,24)(25,27,30,28)(26,31,34,32)(29,35,37,36)(33,38,40,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,21)(13,17)(14,20)(15,19)(18,22)(23,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(9,12)(10,17)(11,21)(13,16)(14,22)(15,24)(18,20)(19,23)(25,28,30,27)(26,32,34,31)(29,35,37,36)(33,38,40,39);(9,13)(10,18)(11,17)(12,19)(14,23)(15,21)(16,22)(20,24)(25,29,30,37)(26,33,34,40)(27,35,28,36)(31,38,32,39)

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,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, 8004241);