#357814

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

2622 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:

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

Polycyclic presentation:

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

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, 265010);