#142959

order	512
nilpotency class	4
isoclinism	isoclinic class of #142933
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	#537	D₄
LCS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

449 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 449 conjugacy classes of subgroups — too many to draw. Only its 56 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=g,e^2=gi,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fhi,c^b=cf,d^b=dg,f^b=fg,h^b=hi,d^c=dh,e^c=egi,f^c=fg,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50155);