#350073

order	512
nilpotency class	3
isoclinism	isoclinic class of #346898
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

810 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)(10,15)(11,20)(13,14)(16,23)(17,24)(21,22)(25,26,30,28)(27,33,34,40)(29,37,39,32)(31,38,35,36);(2,5)(6,8)(9,10)(11,17)(12,15)(13,18)(14,19)(16,21)(20,23)(22,24)(27,34)(31,35)(33,40)(36,38);(1,3,4,7)(2,6,5,8)(9,11,12,21)(10,16,15,17)(13,22,14,20)(18,23,19,24)(25,27)(26,31)(28,35)(29,38)(30,34)(32,40)(33,37)(36,39);(9,11,12,21)(10,17,15,16)(13,20,14,22)(18,23,19,24)(25,28,30,26)(27,35,34,31)(29,32,39,37)(33,38,40,36)

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,c^b=ch,f^c=fh,g^c=gi,f^d=fi,g^d=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 257269);