#152993

order	512
nilpotency class	5
isoclinism	isoclinic class of #13691
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2206
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 ◁ C₂²×C₄ ◁ 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)(5,6,9,15,27,42,59,66,11,16,31,44,28,48,60,68,12,18,32,46,26,14,40,41,10,33,30,63,8,17,25,45)(7,19,21,47,20,53,54,38,22,51,57,35,55,49,62,39,24,52,58,65,56,67,64,36,34,50,61,37,23,43,29,13);(1,2,3,4)(5,7)(6,13)(8,20)(9,29)(10,22)(11,34)(12,24)(14,35)(15,43)(16,36)(17,47)(18,39)(19,45)(21,25)(23,27)(26,55)(28,56)(30,54)(31,64)(32,62)(33,38)(37,42)(40,57)(41,51)(44,67)(46,49)(48,65)(50,66)(52,68)(53,63)(58,60)(59,61);(5,8,10,26,12,28,11,27)(6,14,16,17,18,42,33,48)(7,20,22,55,24,56,34,23)(9,25,30,40,32,60,31,59)(13,35,36,47,39,37,38,65)(15,41,44,45,46,66,63,68)(19,49,50,53,52,43,51,67)(21,54,57,62,58,64,61,29)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=e,d^2=gh,e^2=i,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=ei,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, 60189);