#125424

order	512
nilpotency class	4
isoclinism	isoclinic class of #125422
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×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

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

normal subgroups only

This group has 295 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) = 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)(9,10,14,21,16,24,42,20,17,22,41,56,15,23,13,39)(11,25,28,47,30,60,27,48,31,59,67,18,29,46,43,49)(12,32,35,55,37,19,40,52,38,54,44,51,36,53,34,57)(26,61,63,72,33,58,62,45,65,71,68,50,64,70,66,69);(2,5)(3,8)(6,7)(9,11,17,31)(10,18,22,47)(12,33,38,64)(13,28,42,67)(14,43,41,27)(15,30,16,29)(19,50,53,72)(20,46,39,60)(21,25,56,59)(23,49,24,48)(26,36,65,37)(32,69,54,45)(34,62,40,66)(35,63,44,68)(51,70,55,58)(52,61,57,71);(2,6)(5,7)(9,12,16,37,17,38,15,36)(10,19,23,32,22,53,24,54)(11,26,30,33,31,65,29,64)(13,40,14,44,42,34,41,35)(18,45,48,72,47,69,49,50)(20,55,21,57,39,51,56,52)(25,58,46,61,59,70,60,71)(27,66,67,63,43,62,28,68)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fgi,d^2=gi,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=efg,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, 32620);