#453027

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

771 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:

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

Polycyclic presentation:

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

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, 360223);