#469429

order	512
nilpotency class	3
isoclinism	isoclinic class of #454982
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

909 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:

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

Polycyclic presentation:

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

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, 376625);