#129499

order	512
nilpotency class	3
isoclinism	isoclinic class of #8346
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1779
prime radical	#3449	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

816 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 140 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)(3,5)(4,7)(6,8)(9,10,13,19,14,20,30,42,15,18,31,39,32,40,55,63)(11,16,23,34,24,35,49,59,22,36,46,60,47,61,67,72)(12,25,27,41,28,51,53,64,26,17,29,37,52,38,54,62)(21,33,44,56,45,57,66,70,43,50,48,58,65,69,68,71);(2,5)(3,8)(6,7)(9,11)(10,16)(12,21)(13,23)(14,24)(15,22)(17,33)(18,36)(19,34)(20,35)(25,50)(26,43)(27,44)(28,45)(29,48)(30,49)(31,46)(32,47)(37,56)(38,57)(39,60)(40,61)(41,58)(42,59)(51,69)(52,65)(53,66)(54,68)(55,67)(62,70)(63,72)(64,71);(9,12)(10,17)(11,21)(13,29)(14,28)(15,26)(16,33)(18,25)(19,41)(20,38)(22,43)(23,48)(24,45)(27,31)(30,54)(32,52)(34,58)(35,57)(36,50)(37,39)(40,51)(42,64)(44,46)(47,65)(49,68)(53,55)(56,60)(59,71)(61,69)(62,63)(66,67)(70,72)

Polycyclic presentation:

2^9|a^2=f,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dg,e^b=ei,d^c=di,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36695);