#150877

order	512
nilpotency class	5
isoclinism	isoclinic class of #150788
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	#33600	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 262 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58073);