#405721

order	512
nilpotency class	3
isoclinism	isoclinic class of #394275
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312917);