#143085

order	512
nilpotency class	4
isoclinism	isoclinic class of #143082
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
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

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

normal subgroups only

This group has 373 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)(3,11)(4,9)(5,8)(6,10)(7,12)(13,16)(14,15)(17,18,21,26)(19,28,31,46)(20,34,36,45)(22,27,23,39)(24,40,41,30)(25,37,44,38)(29,47,42,35)(32,43,33,48);(1,3)(2,7)(4,14)(5,12)(6,13)(8,16)(9,11)(10,15)(17,19,21,31)(18,24,26,41)(20,35,36,47)(22,33,23,32)(25,43,44,48)(27,29,39,42)(28,45,46,34)(30,37,40,38);(2,8)(4,5)(7,11)(9,10)(12,14)(15,16)(17,18)(19,29)(20,25)(21,26)(22,39)(23,27)(24,33)(28,35)(30,48)(31,42)(32,41)(34,38)(36,44)(37,45)(40,43)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gi,c^b=cf,d^b=dg,f^b=fi,g^b=gi,d^c=dh,e^c=eh,f^c=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50281);