#150389

order	512
nilpotency class	5
isoclinism	isoclinic class of #150243
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	#2215	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

218 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=fghi,d^2=gh,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gh,h^a=hi,c^b=ci,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,h^b=hi

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, 57585);