#460967

order	512
nilpotency class	3
isoclinism	isoclinic class of #455393
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2700
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16375
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

839 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,d^c=de,g^c=gi,h^c=hi,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, 368163);