#448413

order	512
nilpotency class	3
isoclinism	isoclinic class of #417575
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

996 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=e,d^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^c=fi,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, 355609);