#457879

order	512
nilpotency class	3
isoclinism	isoclinic class of #454372
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
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

857 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=e,c^2=ef,d^2=e,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,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 365075);