#320116

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=eghi,c^2=h,d^2=hi,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, 227312);