#467745

order	512
nilpotency class	3
isoclinism	isoclinic class of #455572
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

834 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=fg,c^2=i,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,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, 374941);