#444392

order	512
nilpotency class	3
isoclinism	isoclinic class of #419852
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=ef,c^2=g,d^2=efi,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=de,e^b=ei,f^b=fi,h^c=hi,e^d=ei,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, 351588);