#487434

order	512
nilpotency class	4
isoclinism	isoclinic class of #486803
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
prime radical	#29686	C₂
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

790 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,27)(20,24)(21,33)(22,26)(23,30)(25,36)(28,42)(29,41)(31,39)(32,46)(34,38)(35,40)(37,43)(44,48)(45,47);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19)(18,23)(20,28)(21,34)(22,29)(24,37)(25,41)(26,38)(27,33)(30,36)(31,44)(32,45)(35,43)(39,47)(40,48)(42,46);(1,2)(3,7)(4,13)(5,8)(6,9)(10,11)(12,16)(14,15)(17,20,22,31)(18,24,26,39)(19,28,29,44)(21,35,36,32)(23,37,38,47)(25,40,33,46)(27,42,41,48)(30,45,34,43);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(19,29)(20,31)(23,38)(24,39)(27,41)(30,34)(32,35)(40,46)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=gi,e^2=g,b^a=be,c^a=cf,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

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318749
maximal normal subgroup of	#72318749

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394630);