#443483

order	512
nilpotency class	3
isoclinism	isoclinic class of #420453
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

1266 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=gi,d^2=e,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^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, 350679);