#455481

order	512
nilpotency class	3
isoclinism	isoclinic class of #454110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1141 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,c^2=e,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi,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, 362677);