#152991

order	512
nilpotency class	5
isoclinism	isoclinic class of #150123
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

326 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=ei,d^2=ghi,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,f^b=fg,g^b=gh,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60187);