#124914

order	512
nilpotency class	4
isoclinism	isoclinic class of #124851
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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

347 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,5)(4,6)(7,8)(9,10,14,21,16,24,41,20,17,22,40,53,15,23,13,38)(11,25,28,45,30,56,27,46,31,55,64,18,29,44,42,47)(12,32,34,48,36,19,33,49,37,51,68,52,35,50,39,66)(26,57,59,65,61,54,63,70,62,72,67,71,60,69,58,43);(2,5)(3,8)(6,7)(9,11,17,31)(10,18,22,45)(12,26,37,62)(13,28,41,64)(14,42,40,27)(15,30,16,29)(19,43,50,70)(20,44,38,56)(21,25,53,55)(23,47,24,46)(32,65,51,71)(33,67,39,59)(34,58,68,63)(35,61,36,60)(48,72,52,57)(49,54,66,69);(2,6)(5,7)(9,12)(10,19,22,50)(11,26)(13,39)(14,34)(15,35)(16,36)(17,37)(18,43,45,70)(20,52,38,48)(21,49,53,66)(23,32,24,51)(25,54,55,69)(27,63)(28,59)(29,60)(30,61)(31,62)(33,41)(40,68)(42,58)(44,57,56,72)(46,71,47,65)(64,67)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fgh,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cfh,d^a=df,f^a=fi,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32110);