#125525

order	512
nilpotency class	4
isoclinism	isoclinic class of #125524
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 237 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)(3,9,11,6)(4,12)(5,8)(7,13)(10,14,16,15)(17,18,21,28,23,29,19,31,24,30,33,25,22,27,34,42)(20,35,36,48,39,26,43,41,37,45,40,44,38,46,32,47);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19,24,34)(18,25,30,28)(20,36,37,40)(21,22,33,23)(26,44,46,48)(27,42,29,31)(32,38,43,39)(35,47,45,41);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,20,22,38,24,37,23,39)(18,26,29,45,30,46,27,35)(19,32,21,40,34,43,33,36)(25,41,42,44,28,47,31,48)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=f,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cfg,d^a=df,f^a=fi,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32721);