#380268

order	512
nilpotency class	3
isoclinism	isoclinic class of #380266
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14262
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1825 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287464);