#460707

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1031 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=ei,d^2=i,f^2=i,g^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,g^c=gi,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, 367903);