#474255

order	512
nilpotency class	3
isoclinism	isoclinic class of #455690
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2825
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16928
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

764 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efgh,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^c=gi,h^c=hi,g^d=gi,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, 381451);