#8103931

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

4571 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=g,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, 8011127);