#8677043

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

4441 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8584239);