#468395

order	512
nilpotency class	3
isoclinism	isoclinic class of #455563
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2730
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16609
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

748 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)(3,13)(4,19)(5,22)(6,25)(7,12)(8,16)(9,20)(10,14)(11,26)(15,28)(17,29)(18,30)(21,31)(23,27)(24,32);(1,3)(2,8,12,29)(4,14)(5,15)(6,16)(7,18)(9,10,28,27)(11,13,25,30)(17,26)(19,22,31,32)(20,24)(21,23);(1,4)(2,9,12,28)(3,14)(5,17)(6,20)(7,21)(8,27,29,10)(11,19,25,31)(13,32,30,22)(15,26)(16,24)(18,23);(1,5,6,14,7,24,26,23)(2,10,11,22,12,27,25,32)(3,15,16,4,18,20,17,21)(8,28,13,31,29,9,30,19)

Polycyclic presentation:

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

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, 375591);