#470567

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

906 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)(4,5)(7,8)(9,10,13,18,14,17,26,31)(11,19,20,32,22,15,25,27)(12,16,24,29,23,30,36,40)(21,28,34,38,35,33,37,39);(2,5)(6,8)(11,20,22,25)(15,27,19,32)(21,34,35,37)(28,38,33,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,21)(13,25)(14,22)(16,28)(17,19)(18,32)(20,26)(23,35)(24,37)(27,31)(29,39)(30,33)(34,36)(38,40);(3,7)(6,8)(9,12)(10,16)(11,21)(13,24)(14,23)(15,28)(17,30)(18,29)(19,33)(20,34)(22,35)(25,37)(26,36)(27,38)(31,40)(32,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de,g^c=gi,h^c=hi,f^d=fi

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, 377763);