#142231

order	512
nilpotency class	4
isoclinism	isoclinic class of #142228
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	#3449	C₂²
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

729 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 729 conjugacy classes of subgroups — too many to draw. Only its 70 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)(4,7)(6,8)(9,10,16,21)(11,22,26,54)(12,27,32,51)(13,19,35,50)(14,36,15,20)(17,39,43,24)(18,31,46,30)(23,55,57,71)(25,60,37,53)(28,52,61,72)(29,62,33,66)(34,49,38,48)(40,67,70,63)(41,58,42,59)(44,56,69,68)(45,65,47,64);(2,5)(3,8)(6,7)(9,11)(10,17)(12,28)(13,24)(14,25)(15,37)(16,26)(18,44)(19,22)(20,41)(21,43)(23,31)(27,40)(29,63)(30,57)(32,61)(33,67)(34,58)(35,39)(36,42)(38,59)(45,71)(46,69)(47,55)(48,53)(49,60)(50,54)(51,70)(52,62)(56,65)(64,68)(66,72);(9,12,16,32)(10,18,21,46)(11,23,26,57)(13,33,35,29)(14,31,15,30)(17,40,43,70)(19,47,50,45)(20,51,36,27)(22,52,54,72)(24,56,39,68)(25,61,37,28)(34,64,38,65)(41,44,42,69)(48,62,49,66)(53,71,60,55)(58,67,59,63)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=g,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,c^b=cf,d^b=dg,f^b=fi,h^b=hi,d^c=dh,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49427);