#507682

order	512
nilpotency class	4
isoclinism	isoclinic class of #501330
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

528 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 130 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 — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=egi,c^2=g,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,f^c=fh,h^c=hi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414878);