#320081

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

762 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)(27,33)(28,31)(29,32)(30,36)(34,38)(35,39)(37,40);(1,3,4,8)(2,5,6,7)(9,11,14,19)(10,16,12,18)(13,20,15,22)(17,23,21,24)(25,27,34,35,28,36,29,37)(26,30,38,40,31,33,32,39);(9,12,14,10)(11,18,19,16)(13,17,15,21)(20,23,22,24)(26,31)(27,35)(29,34)(30,39)(33,40)(36,37);(2,6)(5,7)(11,19)(16,18)(20,22)(23,24)

Polycyclic presentation:

2^9|a^2=fhi,b^2=egh,c^2=h,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, 227277);