#488543

order	512
nilpotency class	4
isoclinism	isoclinic class of #488539
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|d^2=i,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395739);