#425209

order	512
nilpotency class	3
isoclinism	isoclinic class of #417802
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
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

1023 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)(9,10,15,13)(11,19,22,36)(12,23,27,34)(14,29,31,18)(16,32,28,21)(17,30,25,26)(20,33,24,40)(35,38,39,37);(1,3,5,8)(2,6,4,7)(9,11,14,21,15,22,31,32)(10,16,18,19,13,28,29,36)(12,24,26,38,27,20,30,37)(17,33,23,39,25,40,34,35);(1,4,5,2)(3,7,8,6)(9,12,15,27)(10,17,13,25)(11,20,22,24)(14,30,31,26)(16,33,28,40)(18,34,29,23)(19,35,36,39)(21,38,32,37);(3,8)(6,7)(9,13,15,10)(11,19,22,36)(12,25,27,17)(14,18,31,29)(16,32,28,21)(20,35,24,39)(23,30,34,26)(33,37,40,38)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=ef,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,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, 332405);