#320115

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
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

713 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)(9,10,15,21,14,12,13,17)(11,16,22,24,19,18,20,23)(25,26,31,36,30,35,29,34)(27,37,38,33,28,32,39,40);(1,3,4,8)(2,5,6,7)(9,11,14,19)(10,16,12,18)(13,20,15,22)(17,23,21,24)(25,27,29,39,30,28,31,38)(26,32,34,33,35,37,36,40);(9,12,14,10)(11,18,19,16)(13,17,15,21)(20,23,22,24)(25,28,30,27)(26,33,35,40)(29,39,31,38)(32,36,37,34);(2,6)(5,7)(11,19)(16,18)(20,22)(23,24)(25,29,30,31)(26,34,35,36)(27,38,28,39)(32,40,37,33)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 227311);