#469109

order	512
nilpotency class	3
isoclinism	isoclinic class of #455563
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2730
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16645
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

688 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=fg,c^2=g,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,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, 376305);