#152808

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	D₄×C₁₆
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 214 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=d,e^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,e^c=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60004);