#254798

order	512
nilpotency class	3
isoclinism	isoclinic class of #249259
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#36709	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1083 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,d^c=dhi,f^c=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161994);