#421577

order	512
nilpotency class	3
isoclinism	isoclinic class of #418484
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

874 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)(3,5)(4,6)(7,8)(9,10,14,19,15,21,30,20)(11,22,24,32,25,18,13,16)(12,17,28,35,29,36,40,37)(23,38,39,33,26,34,27,31);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,26)(13,14)(15,25)(17,33)(18,19)(20,22)(21,32)(23,29)(24,30)(27,40)(28,39)(31,36)(34,37)(35,38);(2,6)(5,8)(9,12,14,28,15,29,30,40)(10,17,19,35,21,36,20,37)(11,23,24,39,25,26,13,27)(16,31,22,38,32,33,18,34);(9,13,30,25,15,24,14,11)(10,18,20,32,21,22,19,16)(12,27,40,26,29,39,28,23)(17,34,37,33,36,38,35,31)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 328773);