#455702

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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1123 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)(9,10,14,19)(11,20,23,33)(12,24,26,17)(13,16,28,18)(15,27,31,22)(21,35,37,40)(25,32,29,34)(30,36,39,38);(1,2,4,5)(3,6,7,8)(9,11,14,23)(10,15,19,31)(12,21,26,37)(13,27,28,22)(16,20,18,33)(17,30,24,39)(25,38,29,36)(32,35,34,40);(1,3)(2,6)(4,7)(5,8)(10,16)(12,25)(13,28)(15,20)(17,24)(18,19)(21,36)(22,27)(26,29)(31,33)(35,40)(37,38);(3,7)(6,8)(9,12)(10,17)(11,21)(13,29)(14,26)(15,30)(16,32)(18,34)(19,24)(20,35)(22,38)(23,37)(25,28)(27,36)(31,39)(33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=ef,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, 362898);