#129655

order	512
nilpotency class	3
isoclinism	isoclinic class of #8528
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1779
prime radical	#3529	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

575 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 575 conjugacy classes of subgroups — too many to draw. Only its 112 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dgi,e^b=eg,f^b=fi,d^c=dg,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36851);