#513083

order	512
nilpotency class	5
isoclinism	isoclinic class of #512939
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	C₂³	C₂×D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

325 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fg,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^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420279);