#513066

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

308 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 162 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,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, 420262);