#453826

order	512
nilpotency class	3
isoclinism	isoclinic class of #427374
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

882 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=eg,b^2=eh,c^2=i,d^2=ef,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,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, 361022);