#135812

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1850
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

310 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 116 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)(9,10,13,18,14,19,28,38,15,20,29,39,30,40,51,61)(11,16,23,33,24,34,47,57,22,32,45,55,46,56,66,70)(12,17,26,36,27,37,50,60,25,35,48,58,49,59,67,71)(21,41,43,62,44,63,65,72,42,31,52,53,64,54,68,69);(2,5)(3,8)(6,7)(9,11,13,23,14,24,28,47,15,22,29,45,30,46,51,66)(10,16,18,33,19,34,38,57,20,32,39,55,40,56,61,70)(12,21,26,43,27,44,50,65,25,42,48,52,49,64,67,68)(17,31,36,53,37,54,60,69,35,41,58,62,59,63,71,72);(9,12)(10,17)(11,21)(13,26)(14,27)(15,25)(16,31)(18,36)(19,37)(20,35)(22,42)(23,43)(24,44)(28,50)(29,48)(30,49)(32,41)(33,53)(34,54)(38,60)(39,58)(40,59)(45,52)(46,64)(47,65)(51,67)(55,62)(56,63)(57,69)(61,71)(66,68)(70,72)

Polycyclic presentation:

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

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, 43008);