#425101

order	512
nilpotency class	3
isoclinism	isoclinic class of #417732
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1062 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)(9,10,14,19)(11,20,23,31)(12,17,25,33)(13,15,26,18)(16,27,32,21)(22,37,39,35)(24,34,28,29)(30,36,40,38);(1,3)(2,6)(4,7)(5,8)(10,15)(11,21)(12,24)(13,26)(17,33)(18,19)(20,31)(23,27)(25,28)(30,37)(35,40)(36,38);(1,4,5,2)(3,7,8,6)(9,11,14,23)(10,16,19,32)(12,22,25,39)(13,27,26,21)(15,20,18,31)(17,30,33,40)(24,36,28,38)(29,35,34,37);(3,8)(6,7)(9,12,14,25)(10,17,19,33)(11,22,23,39)(13,28,26,24)(15,29,18,34)(16,30,32,40)(20,35,31,37)(21,36,27,38)

Polycyclic presentation:

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

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, 332297);