#149658

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

216 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 256

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

Polycyclic presentation:

2^9|a^2=e,c^2=f,d^2=i,e^2=g,g^2=h,h^2=i,b^a=bd,d^a=df,c^b=ci,d^b=di,e^b=efi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56854);