#489930

order	512
nilpotency class	4
isoclinism	isoclinic class of #489745
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	C₂³	#447
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

640 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 138 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)(3,12)(4,15)(5,11)(6,10)(7,9)(8,13)(14,16)(17,18,24,20)(19,28,32,46)(21,26,23,33)(22,36,39,47)(25,41,29,31)(27,35,34,38)(30,45,40,44)(37,42,48,43);(1,3,7,4)(2,8,9,16)(5,14,6,13)(10,15,11,12)(17,19)(18,25)(20,29)(21,30)(22,37)(23,40)(24,32)(26,42)(27,44)(28,36)(31,38)(33,43)(34,45)(35,41)(39,48)(46,47);(1,4,7,3)(2,8,9,16)(5,13,6,14)(10,15,11,12)(17,20,24,18)(19,29,32,25)(21,26,23,33)(22,27,39,34)(28,41,46,31)(30,42,40,43)(35,47,38,36)(37,44,48,45);(2,9)(8,16)(10,11)(12,15)(17,21,24,23)(18,26,20,33)(19,30,32,40)(22,35,39,38)(25,42,29,43)(27,36,34,47)(28,45,46,44)(31,37,41,48)

Polycyclic presentation:

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

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, 397126);