#482580

order	512
nilpotency class	4
isoclinism	isoclinic class of #482362
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	#2617	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

896 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)(3,11,13,16)(4,9)(5,10)(6,8)(7,12,15,14)(17,18,21,26,22,25,35,40)(19,27,30,41,28,23,36,38)(20,31,32,42,33,24,34,39)(29,37,45,47,44,43,46,48);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,37)(25,27)(26,41)(31,43)(32,45)(33,44)(34,46)(35,36)(38,40)(39,48)(42,47);(2,8)(7,15)(9,10)(11,16)(19,28)(23,27)(29,44)(30,36)(37,43)(38,41)(45,46)(47,48);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)(17,20,22,33)(18,24,25,31)(19,29,28,44)(21,34,35,32)(23,37,27,43)(26,42,40,39)(30,46,36,45)(38,48,41,47)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318670, #72318712, #72318731
maximal normal subgroup of	#72318670, #72318712, #72318731

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389776);