#152424

order	512
nilpotency class	5
isoclinism	isoclinic class of #13669
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#41	D₈
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 263 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) = 256

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

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

Polycyclic presentation:

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

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, 59620);