#150086

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
prime radical	#121	C₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 217 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) = 72
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57282);