#438002

order	512
nilpotency class	3
isoclinism	isoclinic class of #418976
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1141 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 345198);