#405535

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1191 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=e,d^2=i,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, 312731);