#320001

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

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,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, 227197);