#448626

order	512
nilpotency class	3
isoclinism	isoclinic class of #417575
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1198 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)(10,13)(11,15)(17,18,23,29,21,20,22,25)(19,26,30,31,27,24,28,32);(1,3,5,8)(2,6,4,7)(9,10,12,13)(11,16,15,14)(17,19)(18,24)(20,26)(21,27)(22,28)(23,30)(25,31)(29,32);(1,2,5,4)(3,6,8,7)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,24,27,26)(22,25,23,29)(28,32,30,31);(1,4,5,2)(3,6,8,7)(10,13)(14,16)(19,27)(22,23)(24,26)(25,29)

Polycyclic presentation:

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

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, 355822);