#158685

order	512
nilpotency class	2
isoclinism	isoclinic class of #156684
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

2188 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ef,c^2=gh,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 65881);