#488685

order	512
nilpotency class	4
isoclinism	isoclinic class of #488679
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	#62248	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 756 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) = 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)(17,18,24,29)(19,30,34,46)(20,26,23,39)(21,28,35,47)(22,37,36,27)(25,41,45,33)(31,48,40,43)(32,44,38,42);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,19)(18,25)(20,31)(21,33)(22,38)(23,40)(24,34)(26,42)(27,46)(28,43)(29,45)(30,37)(32,36)(35,41)(39,44)(47,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,20,24,23)(18,26,29,39)(19,31,34,40)(21,22,35,36)(25,42,45,44)(27,28,37,47)(30,48,46,43)(32,33,38,41);(2,8)(6,13)(7,14)(12,15)(17,21,23,36,24,35,20,22)(18,27,26,28,29,37,39,47)(19,32,31,33,34,38,40,41)(25,43,44,46,45,48,42,30)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395881);