#473444

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	#16865
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

834 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)(3,7,14,24)(4,8)(5,18)(6,10)(9,17)(11,21)(12,30,31,22)(13,26,32,23)(15,29,19,25)(16,28)(20,27);(1,3,5,19,6,14,21,15)(2,7,11,29,10,24,18,25)(4,13,20,31,17,32,16,12)(8,26,28,30,9,23,27,22);(1,4)(2,8)(3,12)(5,20)(6,17)(7,22)(9,10)(11,28)(13,19)(14,31)(15,32)(16,21)(18,27)(23,25)(24,30)(26,29);(1,4)(2,9,10,8)(3,13)(5,20)(6,17)(7,23,24,26)(11,27,18,28)(12,15)(14,32)(16,21)(19,31)(22,25,30,29)

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,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, 380640);