#513062

order	512
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂²	C₂²×D₁₆
prime radical	C₂³	C₂×D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

308 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234362, #72321181, #72321194, …
maximal normal subgroup of	#72234362, #72321181, #72321194, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420258);