#8505577

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

4521 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)(3,7)(4,6)(5,8)(9,10,15,12)(11,16,21,19)(13,22,24,17)(14,18,20,23)(25,26,31,29)(27,34,32,28)(30,33,39,37)(35,38,40,36);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,19)(13,14)(15,21)(17,23)(18,22)(20,24)(26,29)(28,34)(33,37)(36,38);(9,10,15,12)(11,19,21,16)(13,17,24,22)(14,18,20,23)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,38)(39,40);(1,2,4,6)(3,5,8,7)(9,12,15,10)(11,19,21,16)(13,22,24,17)(14,18,20,23)(25,28,31,34)(26,32,29,27)(30,38,39,36)(33,35,37,40);(1,3)(2,5)(4,8)(6,7)(9,13)(10,17)(11,20)(12,22)(14,21)(15,24)(16,18)(19,23)(25,29,31,26)(27,34,32,28)(30,37,39,33)(35,38,40,36);(1,2)(3,5)(4,6)(7,8)(9,14,15,20)(10,18,12,23)(11,13,21,24)(16,22,19,17)(25,30,31,39)(26,33,29,37)(27,35,32,40)(28,36,34,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=gi,e^2=h,f^2=hi,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, 8412773);