#482608

order	512
nilpotency class	4
isoclinism	isoclinic class of #482397
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×C₂²⋊C₄	C₂⁴
prime radical	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

867 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)(19,28)(20,25,34,42)(21,26,35,23)(22,27)(24,29)(30,40,32,39)(31,36,43,38)(33,41)(37,46)(44,47)(45,48);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(18,23)(19,29)(21,35)(24,38)(25,37)(26,27)(28,36)(30,32)(33,46)(39,48)(40,47)(41,42);(2,8)(7,15)(9,10)(11,16)(17,19,22,29)(18,24,27,28)(20,32,34,30)(21,31,35,43)(23,36,26,38)(25,40,42,39)(33,45,46,44)(37,48,41,47);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)(17,20)(18,25)(19,30)(21,33)(22,34)(23,37)(24,39)(26,41)(27,42)(28,40)(29,32)(31,44)(35,46)(36,47)(38,48)(43,45)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389804);