#138744

order	512
nilpotency class	4
isoclinism	isoclinic class of #12483
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1879
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

339 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=ghi,e^2=hi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=eh,f^c=fh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45940);