#473679

order	512
nilpotency class	3
isoclinism	isoclinic class of #459487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2679
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

976 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)(3,6)(4,7)(5,8)(9,10)(11,19)(12,23)(13,27)(14,18)(15,22)(16,26)(17,28)(20,36)(21,35)(24,32)(25,33)(29,37)(30,39)(31,40)(34,38);(1,3,5,4)(2,6,8,7)(9,11)(10,15,18,30)(12,20)(13,22)(14,21)(16,31,23,29)(17,19,27,35)(24,40)(25,37)(26,38)(28,39)(32,36,33,34);(1,3,5,4)(2,7,8,6)(9,12,13,24,14,26,28,25)(10,16,17,32,18,23,27,33)(11,20,22,40,21,38,39,37)(15,29,19,34,30,31,35,36);(1,4,5,3)(2,7,8,6)(11,21)(12,24,26,25)(15,30)(16,32,23,33)(19,35)(20,37,38,40)(22,39)(29,36,31,34)

Polycyclic presentation:

2^9|b^2=fgh,c^2=ef,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=di,d^c=de,e^c=ei,f^c=fi,h^c=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, 380875);