#8459105

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

4557 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)(3,4)(5,7)(6,8)(12,20)(16,23)(19,24)(21,22)(25,26,32,30)(27,36,38,33)(28,39,34,29)(31,35,40,37);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,27,32,38)(26,33,30,36)(28,31,34,40)(29,35,39,37);(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,24)(22,23)(25,28)(26,29)(27,31)(30,39)(32,34)(33,35)(36,37)(38,40);(2,7)(4,6)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,29,32,39)(26,34,30,28)(27,37,38,35)(31,36,40,33);(1,3)(2,6)(4,7)(5,8)(9,12,14,20)(10,16,18,23)(11,19,17,24)(13,21,15,22)(25,30,32,26)(27,36,38,33)(28,39,34,29)(31,37,40,35);(1,4,5,6)(2,3,7,8)(9,13,14,15)(10,17,18,11)(12,21,20,22)(16,24,23,19)(25,31,32,40)(26,35,30,37)(27,34,38,28)(29,33,39,36)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=h,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

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, 8366301);