#146024

order	512
nilpotency class	4
isoclinism	isoclinic class of #146024
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2076
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

235 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=dg,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=di,e^b=ei,f^b=fh,d^c=di,e^c=eg,f^c=fhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53220);