#482991

order	512
nilpotency class	4
isoclinism	isoclinic class of #482960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

710 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=ghi,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=ci,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi,d^c=dghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390187);