#150953

order	512
nilpotency class	5
isoclinism	isoclinic class of #7418
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂²	C₂×#357
prime radical	C₂³	#357
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

200 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gh,e^2=fgh,g^2=hi,h^2=i,b^a=bd,d^a=df,d^b=dg,e^b=efg,g^b=gh,h^b=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58149);