#452651

order	512
nilpotency class	3
isoclinism	isoclinic class of #417901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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	#2684
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

810 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359847);