#488852

order	512
nilpotency class	4
isoclinism	isoclinic class of #488707
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
prime radical	#62384	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

659 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 659 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396048);