#144134

order	512
nilpotency class	4
isoclinism	isoclinic class of #144131
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
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

495 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 256

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=gi,e^2=h,f^2=ghi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fghi,g^a=gi,c^b=cf,d^b=dg,e^b=eh,f^b=fgh,g^b=gi,d^c=dg,e^c=eh,f^c=fgh,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, 51330);