#487636

order	512
nilpotency class	4
isoclinism	isoclinic class of #487635
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×C₂²⋊C₄	C₂⁴
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

994 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)(3,12)(4,7)(5,10)(6,11)(8,14)(9,16)(13,27)(15,22)(17,20)(18,25)(19,30)(21,31)(23,28)(24,32)(26,29);(2,7)(3,13)(4,16)(8,21)(9,10)(11,20)(12,26)(14,28)(15,31)(17,32)(19,23)(22,27)(24,25)(29,30);(3,14)(4,17)(8,12)(13,28)(15,31)(16,32)(19,22)(21,26)(23,27)(29,30);(1,3,6,13)(2,8,11,23)(4,15,17,29)(5,14,18,28)(7,19,20,21)(9,22,24,26)(10,12,25,27)(16,30,32,31)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338803, #72338813, #72338817
maximal normal subgroup of	#72338803, #72338813, #72338817

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394832);