#474241

order	512
nilpotency class	3
isoclinism	isoclinic class of #455690
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2825
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16920
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

631 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=egh,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,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, 381437);