#489604

order	512
nilpotency class	4
isoclinism	isoclinic class of #489394
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	#62263	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

724 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)(3,11)(4,10)(5,8)(6,9)(7,13)(12,15)(14,16)(17,18,22,26)(19,27,29,45)(20,31,33,47)(21,36,34,25)(23,38,39,28)(24,41,43,35)(30,46,37,42)(32,44,48,40);(1,3)(2,7)(4,14)(5,13)(6,12)(8,15)(9,16)(10,11)(17,19)(18,23)(20,32)(21,37)(22,29)(24,42)(25,44)(26,39)(27,31)(28,35)(30,34)(33,48)(36,40)(38,41)(43,46)(45,47);(2,8)(4,5)(7,15)(9,10)(11,16)(13,14)(20,33)(24,43)(27,45)(28,38)(31,47)(32,48)(35,41)(42,46);(2,9)(3,12)(8,10)(13,14)(17,20,34,41,22,33,21,35)(18,24,36,47,26,43,25,31)(19,28,37,48,29,38,30,32)(23,27,44,46,39,45,40,42)

Polycyclic presentation:

2^9|a^2=i,d^2=gh,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,d^c=di,e^c=eh,f^c=fhi

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, 396800);