#8550975

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

4526 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8458171);