#146025

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:

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

Polycyclic presentation:

2^9|a^2=ei,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, 53221);