#150242

order	512
nilpotency class	5
isoclinism	isoclinic class of #7418
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×#357
prime radical	#33600	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

318 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 84 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)(11,24,27,57)(12,18)(13,35)(14,40)(15,42)(16,23)(17,26,45,60)(19,37)(20,32)(21,33)(22,34)(25,56,30,65)(28,63,29,64)(31,66)(36,53,61,54)(38,52)(39,50)(41,58,43,49)(44,59,55,62)(46,67,47,72)(48,68)(51,69)(70,71);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,25)(13,36)(14,41)(15,43)(16,27)(18,44)(19,49)(20,53)(21,54)(22,55)(23,45)(24,50)(26,38)(28,32)(29,33)(30,34)(31,67)(35,57)(37,60)(39,61)(40,46)(42,47)(48,64)(51,63)(52,58)(56,71)(59,69)(62,68)(65,66)(70,72);(9,12,16,34)(10,18,23,22)(11,25,27,30)(13,31,39,70)(14,32,15,33)(17,44,45,55)(19,48,52,51)(20,42,21,40)(24,56,57,65)(26,59,60,62)(28,43,29,41)(35,66,50,71)(36,67,61,72)(37,68,38,69)(46,53,47,54)(49,64,58,63)

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=gh,f^2=hi,g^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gh,h^a=hi,d^b=dg,e^b=efgh,f^b=fh,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, 57438);