#468363

order	512
nilpotency class	3
isoclinism	isoclinic class of #454025
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1014 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 375559);