#160412

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

Polycyclic presentation:

2^9|a^2=efgi,b^2=efg,c^2=efh,d^2=g,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, 67608);