#466141

order	512
nilpotency class	3
isoclinism	isoclinic class of #456501
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	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1048 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)(6,8)(9,10)(11,20)(12,25)(13,17)(14,19)(15,22)(16,28)(18,30)(21,31)(23,32)(24,33)(26,40)(27,35)(29,34)(36,38)(37,39);(1,2,4,5)(3,6,7,8)(9,11)(10,15,19,33)(12,26)(13,22)(14,23)(16,31,35,40)(17,32,18,20)(21,29)(24,30)(25,39,34,36)(27,38)(28,37);(3,7)(6,8)(9,12)(10,16)(11,21)(13,27)(14,29)(15,31)(17,25)(18,34)(19,35)(20,36)(22,37)(23,26)(24,38)(28,30)(32,39)(33,40);(1,3,4,7)(2,6,5,8)(10,17)(12,27)(13,30)(15,32)(18,19)(20,33)(21,37)(22,24)(25,34)(26,38)(28,29)(36,39)

Polycyclic presentation:

2^9|b^2=efg,d^2=efi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=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, 373337);