#125909

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

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

normal subgroups only

This group has 314 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)(17,18,22,28)(19,30,33,43)(20,29,23,26)(21,25,37,42)(24,34,40,31)(27,35,41,36)(32,44,48,39)(38,46,45,47);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,24)(20,34)(21,32)(22,33)(23,31)(25,38)(26,43)(27,44)(28,40)(29,30)(35,47)(36,46)(37,48)(39,41)(42,45);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,25)(20,23)(21,36)(24,38)(26,41)(27,29)(28,42)(30,44)(31,34)(32,46)(35,37)(39,43)(40,45)(47,48)

Polycyclic presentation:

2^9|a^2=f,c^2=fi,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, 33105);