#436991

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1006 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 344187);