#403035

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

1300 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)(9,10,13,17,15,20,14,19)(11,21,23,16,25,32,24,31)(12,26,27,34,29,33,28,18)(22,30,36,35,38,40,37,39);(1,3,5,4)(2,6,7,8)(9,11)(10,16,20,31)(12,22)(13,23)(14,24)(15,25)(17,32,19,21)(18,30,34,40)(26,35,33,39)(27,36)(28,37)(29,38);(10,17)(13,14)(16,21)(18,33)(19,20)(23,24)(26,34)(27,28)(30,35)(31,32)(36,37)(39,40);(1,4,5,3)(2,6,7,8)(9,12,15,29)(10,18)(11,22,25,38)(13,28,14,27)(16,30)(17,33)(19,26)(20,34)(21,35)(23,37,24,36)(31,40)(32,39)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,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, 310231);