#8505365

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

4411 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=hi,e^2=h,f^2=gi,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, 8412561);