#475063

order	512
nilpotency class	3
isoclinism	isoclinic class of #455107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23428	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1144 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)(3,6)(4,5)(7,8)(17,18,21,27,23,28,22,25)(19,29,30,24,31,26,20,32);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,20,23,31,21,30)(18,24,27,26,28,32,25,29);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(18,25)(19,20)(21,22)(26,29)(27,28)(30,31);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(17,20,21,19,23,30,22,31)(18,26,25,24,28,29,27,32)

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=g,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,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, 382259);