#487421

order	512
nilpotency class	4
isoclinism	isoclinic class of #486891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

938 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)(3,13)(4,11)(5,23)(6,10)(7,14)(8,16)(9,18)(12,17)(15,25)(19,31)(20,28)(21,29)(22,27)(24,32)(26,30);(1,3,7,19)(2,9,14,11)(4,17,18,5)(6,25,26,8)(10,29,30,32)(12,31,23,13)(15,27,16,28)(20,24,22,21);(2,10)(3,6)(4,20)(5,17)(9,12)(11,23)(13,16)(14,30)(15,31)(18,22)(19,26)(21,24)(27,32)(28,29);(1,4,7,18)(2,11,14,9)(3,17,19,5)(6,21,26,24)(8,22,25,20)(10,29,30,32)(12,31,23,13)(15,28,16,27)

Polycyclic presentation:

2^9|b^2=g,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394617);