#300451

order	512
nilpotency class	3
isoclinism	isoclinic class of #300437
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1034 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=f,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207647);