#145212

order	512
nilpotency class	4
isoclinism	isoclinic class of #145211
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2033
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

275 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 275 conjugacy classes of subgroups — too many to draw. Only its 64 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=d,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fi,e^c=egh,f^c=fgi,h^c=hi

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, 52408);