#403037

order	512
nilpotency class	3
isoclinism	isoclinic class of #396101
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1238 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)(9,10,14,34,16,21,15,20)(11,22,25,17,27,38,26,37)(12,28,29,18,24,36,30,39)(13,31,32,40,23,35,33,19);(1,3,5,4)(2,6,7,8)(9,11)(10,17,21,37)(12,23)(13,24)(14,25)(15,26)(16,27)(18,31,39,35)(19,36,40,28)(20,22,34,38)(29,33)(30,32);(9,12,16,24)(10,18,21,39)(11,23,27,13)(14,30,15,29)(17,35,37,31)(19,38,40,22)(20,36,34,28)(25,32,26,33);(1,4,5,3)(2,6,7,8)(9,13,16,23)(10,19)(11,24,27,12)(14,33,15,32)(17,36)(18,38)(20,31)(21,40)(22,39)(25,29,26,30)(28,37)(34,35)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=i,d^2=efi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,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, 310233);