#487808

order	512
nilpotency class	4
isoclinism	isoclinic class of #487643
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	#2021	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

752 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,41)(29,42)(31,45)(32,38)(33,39)(34,40)(35,43)(36,44)(37,46)(47,48);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,22,28)(18,23,26,35)(20,31,32,29)(21,30,34,43)(24,37,38,36)(25,27,40,41)(33,46,47,44)(39,45,48,42);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(19,28)(20,32)(21,34)(23,35)(24,38)(27,41)(39,48)(44,46);(2,7)(6,15)(8,12)(9,16)(17,20,22,32)(18,24,26,38)(19,29,28,31)(21,33,34,47)(23,36,35,37)(25,39,40,48)(27,42,41,45)(30,44,43,46)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395004);