#463957

order	512
nilpotency class	3
isoclinism	isoclinic class of #455393
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2700
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16498
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

677 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)(3,15,19,9)(4,20,21,32)(5,22)(6,13)(7,14)(8,12)(10,17,27,26)(11,24)(16,29)(18,30,25,28)(23,31);(1,3,6,18,8,19,7,25)(2,9,13,28,12,15,14,30)(4,5,17,16,21,24,26,23)(10,29,20,11,27,31,32,22);(1,4,6,17,8,21,7,26)(2,10,14,32,12,27,13,20)(3,16,18,24,19,23,25,5)(9,11,30,29,15,22,28,31);(1,5,6,16,8,24,7,23)(2,11,14,29,12,22,13,31)(3,17,18,21,19,26,25,4)(9,27,30,20,15,10,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371153);