#512981

order	512
nilpotency class	5
isoclinism	isoclinic class of #512827
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#33600	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

377 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 130 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 — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fg,g^a=gh,h^a=hi,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420177);