#455701

order	512
nilpotency class	3
isoclinism	isoclinic class of #453999
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1131 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)(10,15)(11,19)(13,25)(17,34)(18,26)(20,37)(21,27)(22,38)(23,28)(29,32)(30,36)(31,40);(1,2,4,5)(3,6,7,8)(9,10,14,18)(11,16,21,33)(12,17,24,36)(13,26,25,15)(19,29,27,32)(20,31,38,40)(22,39,37,35)(23,30,28,34);(1,3)(2,6)(4,7)(5,8)(9,11,14,21)(10,16,18,33)(12,22,24,37)(13,27,25,19)(15,29,26,32)(17,35,36,39)(20,28,38,23)(30,40,34,31);(3,7)(6,8)(9,12)(10,17)(11,20)(13,28)(14,24)(15,30)(16,31)(18,36)(19,37)(21,38)(22,27)(23,25)(26,34)(29,35)(32,39)(33,40)

Polycyclic presentation:

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

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, 362897);