#143100

order	512
nilpotency class	4
isoclinism	isoclinic class of #143098
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	#28174	C₂
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

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

normal subgroups only

This group has 386 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) = 128

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

Polycyclic presentation:

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

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, 50296);