#472805

order	512
nilpotency class	3
isoclinism	isoclinic class of #456381
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2712
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16837
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

897 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)(3,15)(4,19)(5,16)(6,13)(7,12)(8,14)(9,17)(10,21)(11,26)(18,23)(20,28)(22,29)(24,32)(25,31)(27,30);(1,3)(2,9,14,15)(4,11,22,31)(5,10)(6,18)(7,28)(8,17)(12,23,13,20)(16,21,32,27)(19,25)(24,30)(26,29);(1,4)(2,10)(3,16)(5,23)(6,27)(7,21)(8,22)(9,25)(11,28)(12,29)(13,19)(14,30)(15,26)(17,32)(18,31)(20,24);(1,5,6,25,8,24,7,26)(2,11,12,32,14,31,13,16)(3,10,18,19,17,30,28,29)(4,20,21,9,22,23,27,15)

Polycyclic presentation:

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

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, 380001);