#285907

order	512
nilpotency class	3
isoclinism	isoclinic class of #281787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

610 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)(3,6)(4,5)(7,8)(9,10,12,16,13,17,21,15)(11,14,20,23,19,22,18,24)(25,26,29,33,28,32,35,31)(27,34,37,40,36,30,38,39);(2,5)(6,8)(9,11,12,20,13,19,21,18)(10,14,16,23,17,22,15,24)(25,27,35,38,28,36,29,37)(26,30,31,40,32,34,33,39);(1,3,4,7)(2,6,5,8)(10,15)(11,18)(12,21)(14,22)(16,17)(19,20)(26,31)(27,36)(29,35)(30,39)(32,33)(34,40);(2,5)(6,8)(11,19)(14,22)(18,20)(23,24)(26,32)(27,36)(31,33)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 193103);