#129322

order	512
nilpotency class	3
isoclinism	isoclinic class of #8453
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	C₂³×C₈	C₂³
socle	C₂²	#1779
prime radical	#3449	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

818 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 122 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)(9,10,14,20,15,21,35,48,16,22,36,49,13,19,33,46)(11,23,27,51,25,39,57,47,26,44,34,62,31,17,37,40)(12,18,29,42,28,41,58,67,30,43,60,69,32,45,61,70)(24,52,55,72,56,38,64,66,54,50,59,71,53,65,63,68);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,24)(13,31)(14,34)(15,25)(16,26)(18,38)(19,44)(20,47)(21,23)(22,39)(27,36)(28,56)(29,59)(30,54)(32,53)(33,57)(35,37)(40,49)(41,50)(42,68)(43,65)(45,52)(46,51)(48,62)(55,60)(58,63)(61,64)(66,69)(67,72)(70,71);(9,12,15,28,16,30,13,32)(10,18,21,41,22,43,19,45)(11,24,25,56,26,54,31,53)(14,29,35,58,36,60,33,61)(17,38,23,50,39,65,44,52)(20,42,48,67,49,69,46,70)(27,55,57,64,34,59,37,63)(40,66,51,71,47,68,62,72)

Polycyclic presentation:

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

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, 36518);