#487316

order	512
nilpotency class	4
isoclinism	isoclinic class of #486819
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×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

841 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,8)(4,9)(5,14)(6,10)(7,11)(12,15)(13,16)(17,18)(19,27)(20,24)(21,34)(22,26)(23,29)(25,37)(28,44)(30,43)(31,47)(32,41)(33,48)(35,39)(36,42)(38,45)(40,46);(1,3)(2,7)(4,11)(5,13)(6,12)(8,14)(9,15)(10,16)(17,19,22,30)(18,23,26,39)(20,31,32,28)(21,35,37,29)(24,40,41,38)(25,43,34,27)(33,45,36,46)(42,44,48,47);(1,3)(2,8)(4,13)(5,11)(6,12)(7,14)(9,16)(10,15)(17,20)(18,24)(19,28)(21,36)(22,32)(23,38)(25,42)(26,41)(27,44)(29,46)(30,31)(33,37)(34,48)(35,45)(39,40)(43,47);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,15,16,8)(20,32)(24,41)(28,31)(33,36)(38,40)(42,48)(44,47)(45,46)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341031
maximal normal subgroup of	#72341031

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394512);