#362411

order	512
nilpotency class	3
isoclinism	isoclinic class of #360368
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2344
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13914
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

695 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269607);