#403033

order	512
nilpotency class	3
isoclinism	isoclinic class of #396094
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1408 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310229);