#146260

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

265 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=dgh,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, 53456);