#8092474

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
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

4623 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)(9,10,13,14)(11,15,17,21)(12,19,20,16)(18,24,23,22)(25,26,30,32)(27,35,36,31)(28,33,29,34)(37,40,38,39);(3,7)(5,8)(10,14)(15,21)(16,19)(22,24)(25,27)(26,31)(28,37)(29,38)(30,36)(32,35)(33,39)(34,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(26,32)(31,35)(33,34)(39,40);(3,7)(5,8)(10,14)(11,17)(16,19)(18,23)(26,32)(27,36)(33,34)(37,38);(2,6)(5,8)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,28,30,29)(26,33,32,34)(27,37,36,38)(31,39,35,40);(25,29,30,28)(26,34,32,33)(27,38,36,37)(31,40,35,39)

Polycyclic presentation:

2^9|a^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,d^c=dghi

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, 7999670);