#461463

order	512
nilpotency class	3
isoclinism	isoclinic class of #454981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2619
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16392
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

933 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,h^c=hi,f^d=fi,h^d=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, 368659);