#8105901

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
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

4576 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)(4,6)(7,8)(9,10)(11,18)(12,21)(13,17)(14,16)(15,20)(19,23)(22,24)(25,26)(27,33)(28,32)(29,31)(30,34)(35,40)(36,38)(37,39);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,22)(13,19)(14,20)(16,18)(17,24)(21,23)(25,27)(26,30)(28,36)(29,34)(31,33)(32,40)(35,37)(38,39);(1,3,4,8)(2,5,6,7)(26,31)(30,33)(32,39)(38,40);(10,16)(15,18)(17,21)(23,24)(26,31)(27,34)(32,39)(35,36);(2,6)(5,7)(9,12,14,13)(10,17,16,21)(11,19,20,22)(15,23,18,24)(25,28,29,37)(26,32,31,39)(27,35,34,36)(30,38,33,40);(9,13,14,12)(10,17,16,21)(11,19,20,22)(15,24,18,23)

Polycyclic presentation:

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

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, 8013097);