#443497

order	512
nilpotency class	3
isoclinism	isoclinic class of #420461
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1272 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)(11,21)(12,25)(13,19)(14,20)(15,18)(16,23)(17,27)(22,36)(24,31)(26,35)(28,34)(29,33)(30,39)(32,38)(37,40);(1,2,4,5)(3,7,8,6)(9,11,13,23)(10,16,19,21)(12,22,26,37)(14,28,15,24)(17,32,33,30)(18,34,20,31)(25,40,35,36)(27,39,29,38);(9,12)(10,17,19,33)(11,22)(13,26)(14,27)(15,29)(16,30,21,32)(18,25,20,35)(23,37)(24,38)(28,39)(31,40,34,36);(1,3,4,8)(2,6,5,7)(10,18)(11,23)(14,15)(16,31)(17,25)(19,20)(21,34)(22,37)(27,29)(30,40)(32,36)(33,35)

Polycyclic presentation:

2^9|b^2=e,c^2=g,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^c=hi,f^d=fi,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, 350693);