#160372

order	512
nilpotency class	2
isoclinism	isoclinic class of #158815
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2036 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fh,c^2=eg,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de

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, 67568);