#453851

order	512
nilpotency class	3
isoclinism	isoclinic class of #417902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

644 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)(9,10,14,18,15,20,13,19)(11,21,24,16,25,32,23,33)(12,26,30,17,28,36,29,35)(22,37,38,34,39,31,27,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,27)(13,23)(14,24)(15,25)(17,34)(18,32)(19,21)(20,33)(22,30)(26,37)(28,38)(29,39)(31,36)(35,40);(1,2,5,4)(3,6,8,7)(9,12,13,29,15,28,14,30)(10,17,18,36,20,35,19,26)(11,22,24,38,25,39,23,27)(16,31,21,34,33,37,32,40);(1,4,5,2)(3,6,8,7)(11,23,25,24)(12,28)(16,32,33,21)(17,35)(22,38,39,27)(26,36)(29,30)(31,34,37,40)

Polycyclic presentation:

2^9|a^2=efg,b^2=efh,c^2=e,d^2=ef,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,g^b=gi,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, 361047);