#424211

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1563 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=efi,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^d=hi

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, 331407);