#449882

order	512
nilpotency class	3
isoclinism	isoclinic class of #427311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2471
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15952
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1013 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357078);