#151986

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₃₂	D₄
commutator	C₄	C₂×C₄×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	D₄×C₁₆
prime radical	C₂³	#504
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₂ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

218 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59182);