#129281

order	512
nilpotency class	3
isoclinism	isoclinic class of #8465
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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

628 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 80
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 80 points:

(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,22,28,23,29,43,56,24,30,44,57,21,27,41,54)(19,25,33,47,32,46,63,74,34,48,65,76,39,52,69,78)(20,35,38,67,37,50,68,55,36,53,42,70,40,26,49,51)(31,58,61,79,62,45,71,73,59,66,64,80,60,72,77,75);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,25)(20,31)(21,39)(22,33)(23,32)(24,34)(26,45)(27,52)(28,47)(29,46)(30,48)(35,66)(36,59)(37,62)(38,61)(40,60)(41,69)(42,64)(43,63)(44,65)(49,77)(50,72)(51,73)(53,58)(54,78)(55,75)(56,74)(57,76)(67,80)(68,71)(70,79);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,26)(19,31)(21,40)(22,42)(23,37)(24,36)(25,45)(27,53)(28,55)(29,35)(30,50)(32,62)(33,64)(34,59)(38,44)(39,60)(41,68)(43,49)(46,66)(47,75)(48,72)(51,57)(52,58)(54,67)(56,70)(61,65)(63,77)(69,71)(73,76)(74,79)(78,80)

Polycyclic presentation:

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

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, 36477);