#473341

order	512
nilpotency class	3
isoclinism	isoclinic class of #457096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

967 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=efgh,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,h^c=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, 380537);