#403674

order	512
nilpotency class	3
isoclinism	isoclinic class of #395187
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
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

932 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egi,c^2=g,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,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, 310870);