#130458

order	512
nilpotency class	4
isoclinism	isoclinic class of #130252
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3259	C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

517 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=hi,e^2=i,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=eg,f^b=fgh,h^b=hi,d^c=dg,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37654);