#146077

order	512
nilpotency class	4
isoclinism	isoclinic class of #146043
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2077
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#468
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

198 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 70 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)(3,6)(4,7)(5,8)(9,16)(10,14)(11,13)(12,15)(17,18,25,40,21,30,39,42,24,31,38,46,23,28,22,29)(19,32,41,47,36,45,37,48,34,44,35,43,20,26,33,27);(1,3,4,10)(2,6,7,14)(5,9,12,11)(8,16,15,13)(17,19,24,34)(18,26,31,45)(20,21,36,23)(22,33,39,37)(25,41,38,35)(27,46,48,40)(28,44,30,32)(29,43,42,47);(2,7)(3,9)(5,12)(6,13)(10,11)(14,16)(17,20,21,34,24,36,23,19)(18,27,28,47,31,48,30,43)(22,37,25,41,39,33,38,35)(26,42,32,40,45,29,44,46)

Polycyclic presentation:

2^9|a^2=egh,b^2=h,c^2=d,d^2=i,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=di,f^b=fh,e^c=eg,f^c=fhi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53273);