#474254

order	512
nilpotency class	3
isoclinism	isoclinic class of #455822
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

795 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381450);