#489672

order	512
nilpotency class	4
isoclinism	isoclinic class of #489385
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	C₄×C₈	C₂⁴
prime radical	#466	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

633 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=ghi,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi

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, 396868);