#364232

order	512
nilpotency class	3
isoclinism	isoclinic class of #363904
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2297
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13961
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

881 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)(3,12,16,27)(4,17,20,9)(5,21,19,31)(6,10)(7,11)(8,15,24,29)(13,26)(14,28)(18,32)(22,25)(23,30);(1,3,7,16)(2,8)(4,18,20,13)(5,14,19,30)(6,15,22,29)(9,23)(10,12)(11,24)(17,28)(21,26)(25,27)(31,32);(1,4,6,19,7,20,22,5)(2,9,10,21,11,17,25,31)(3,13,15,14,16,18,29,30)(8,23,12,26,24,28,27,32);(1,5,22,20,7,19,6,4)(2,9,10,21,11,17,25,31)(3,14,29,13,16,30,15,18)(8,23,12,26,24,28,27,32)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=e,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^b=hi,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, 271428);