#513208

order	512
nilpotency class	5
isoclinism	isoclinic class of #513173
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	#33846	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

361 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=ei,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,c^b=ci,d^b=de,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420404);