#513276

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	#2246	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

311 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234341
maximal normal subgroup of	#72234341

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420472);