#146227

order	512
nilpotency class	4
isoclinism	isoclinic class of #146221
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2092
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

281 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 281 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=dg,b^2=de,d^2=g,e^2=gi,f^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,d^b=dg,e^b=egi,f^b=fh,g^b=gi,d^c=dg,e^c=eg,f^c=fhi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53423);