#8471327

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

4735 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8378523);