#8105159

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4608 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)(3,5)(4,7)(6,8)(9,10)(11,19)(12,13)(14,18)(15,17)(16,22)(20,21)(23,24)(25,26,30,29)(27,31,35,34)(28,32,37,38)(33,39,36,40);(1,2,5,3)(4,7,8,6)(9,11,15,22)(10,16,17,19)(12,20,18,24)(13,23,14,21)(25,27,30,35)(26,31,29,34)(28,36,37,33)(32,40,38,39);(1,3,5,2)(4,7,8,6)(9,12,15,18)(10,13,17,14)(11,20,22,24)(16,23,19,21)(25,28,30,37)(26,32,29,38)(27,33,35,36)(31,39,34,40);(2,3)(6,7)(10,17)(12,18)(16,19)(20,24)(25,29,30,26)(27,34,35,31)(28,32,37,38)(33,39,36,40);(1,4,5,8)(2,6,3,7)(9,13,15,14)(10,12,17,18)(11,21,22,23)(16,24,19,20)(27,35)(31,34)(33,36)(39,40);(9,14,15,13)(10,18,17,12)(11,21,22,23)(16,24,19,20)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gi,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8012355);