#357923

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,d^c=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265119);