#149601

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
prime radical	C₂⁴	C₂×C₁₆
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

265 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,7,8,5)(4,6)(9,10,12,17,13,18,25,35,14,19,26,36,28,38,47,57,15,20,27,37,29,39,48,58,30,40,49,59,50,60,65,70)(11,16,22,32,23,33,44,54,24,34,45,55,46,56,64,69,21,31,41,51,42,52,61,66,43,53,62,67,63,68,71,72);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,22)(13,23)(14,24)(15,21)(17,32)(18,33)(19,34)(20,31)(25,44)(26,45)(27,41)(28,46)(29,42)(30,43)(35,54)(36,55)(37,51)(38,56)(39,52)(40,53)(47,64)(48,61)(49,62)(50,63)(57,69)(58,66)(59,67)(60,68)(65,71)(70,72);(11,21)(16,31)(22,41)(23,42)(24,43)(32,51)(33,52)(34,53)(44,61)(45,62)(46,63)(54,66)(55,67)(56,68)(64,71)(69,72)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56797);