#462523

order	512
nilpotency class	3
isoclinism	isoclinic class of #455393
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2700
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16435
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

732 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=ef,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, 369719);