#319668

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

727 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 226864);