#437764

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	#23443	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1346 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)(20,34)(27,38)(33,40)(35,37);(1,2)(3,6)(4,5)(7,8)(9,10,13,17,14,18,26,32)(11,19,21,28,22,31,25,15)(12,16,24,30,23,29,36,39)(20,33,35,38,34,40,37,27);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,20)(13,25)(14,22)(16,27)(17,31)(18,28)(19,32)(21,26)(23,34)(24,37)(29,38)(30,40)(33,39)(35,36);(2,5)(6,8)(9,12)(10,16)(11,20)(13,24)(14,23)(15,27)(17,30)(18,29)(19,33)(21,35)(22,34)(25,37)(26,36)(28,38)(31,40)(32,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 344960);