#466138

order	512
nilpotency class	3
isoclinism	isoclinic class of #456505
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1081 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)(11,21)(12,26)(13,19)(14,18)(15,20)(16,23)(17,28)(22,37)(24,33)(25,32)(27,31)(29,35)(30,34)(36,38)(39,40);(1,2,4,5)(3,6,7,8)(9,11,15,25)(10,16)(12,27,30,22)(13,23,14,24)(17,31)(18,21)(19,32)(20,33)(26,36)(28,39,29,38)(34,40)(35,37);(3,7)(6,8)(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,31)(18,26)(19,34)(20,35)(21,36)(23,38)(24,39)(25,27)(32,40)(33,37);(1,3,4,7)(2,6,5,8)(12,28)(13,14)(17,26)(18,19)(21,32)(22,38)(23,24)(27,39)(29,30)(31,36)(34,35)(37,40)

Polycyclic presentation:

2^9|b^2=efg,d^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,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi

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, 373334);