#474687

order	512
nilpotency class	3
isoclinism	isoclinic class of #455107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

782 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=egh,c^2=eg,d^2=ei,e^2=i,g^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,d^c=de,e^c=ei,g^c=gi,h^c=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, 381883);