#414278

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

784 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321474);