#453911

order	512
nilpotency class	3
isoclinism	isoclinic class of #425091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

681 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:

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

Polycyclic presentation:

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

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, 361107);