#479735

order	512
nilpotency class	3
isoclinism	isoclinic class of #476443
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2829
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17231
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

757 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)(3,11)(4,16,20,8)(5,21)(6,10)(7,14)(9,23)(12,28)(13,30,32,25)(15,27)(17,29,31,24)(18,22,26,19);(1,3)(2,7,10,12)(4,17)(5,14)(6,15)(8,25,16,30)(9,11,21,27)(13,19)(18,24,26,29)(20,31)(22,32)(23,28);(1,2)(3,12)(4,18)(5,21)(6,10)(7,15)(8,19)(9,23)(11,14)(13,25)(16,22)(17,24)(20,26)(27,28)(29,31)(30,32);(1,4)(2,8)(3,13)(5,22)(6,20)(7,24)(9,26)(10,16)(11,25)(12,29)(14,17)(15,32)(18,21)(19,23)(27,30)(28,31)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386931);