#125910

order	512
nilpotency class	4
isoclinism	isoclinic class of #125644
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

303 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 303 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=f,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gi,h^a=hi,d^b=df,e^b=egi,d^c=dg,e^c=eh,h^c=hi,e^d=ei

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, 33106);