#490417

order	512
nilpotency class	4
isoclinism	isoclinic class of #490133
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 ◁ C₂×#446 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

656 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=gh,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dghi,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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397613);