#424210

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1515 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)(11,15)(12,26)(13,18)(14,19)(16,27)(17,29)(20,30)(21,37)(22,39)(23,32)(24,33)(25,36)(28,35)(31,38)(34,40);(6,7)(9,11,13,23)(10,15,18,32)(12,22,27,40)(14,25,30,24)(16,34,26,39)(17,31,35,37)(19,36,20,33)(21,29,38,28);(1,2,3,4)(5,6,8,7)(9,12)(10,16)(11,21)(13,27)(14,29)(15,31)(17,20)(18,26)(19,35)(22,25)(23,38)(24,40)(28,30)(32,37)(33,39)(34,36);(5,7,8,6)(9,12)(10,17)(11,22)(13,27)(14,28)(15,31)(16,20)(18,35)(19,26)(21,25)(23,40)(24,38)(29,30)(32,37)(33,34)(36,39)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=efi,d^2=e,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, 331406);