#125429

order	512
nilpotency class	4
isoclinism	isoclinic class of #125422
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1505
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

237 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 66 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)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18,21,24,22,28,37,26,23,27,39,41,31,46,20,35)(19,29,33,43,30,45,36,40,34,47,38,42,48,25,32,44);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(18,24,27,41)(19,30)(20,21)(22,31)(25,42,45,43)(26,28,35,46)(29,40,47,44)(32,36)(34,48)(37,39);(2,7)(6,15)(8,12)(9,16)(17,19,31,48,23,34,22,30)(18,25,28,29,27,45,46,47)(20,36,39,33,37,32,21,38)(24,40,26,42,41,44,35,43)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=fgi,d^2=g,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fi,c^b=ch,d^b=dg,e^b=efgi,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=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, 32625);