#135776

order	512
nilpotency class	4
isoclinism	isoclinic class of #135766
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1846
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#495
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

218 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 106 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)(4,6)(5,13)(7,11)(8,12)(10,15)(14,16)(17,18)(19,29)(20,35,37,25)(21,39)(22,43)(23,27)(24,32)(26,41)(28,40)(30,50)(31,57,59,51)(33,64)(34,53)(36,68,70,72)(38,49,69,67)(42,44)(45,58,65,79)(46,63)(47,55)(48,71,66,56)(52,62)(54,61)(60,75,78,77)(73,76,74,80);(2,6)(7,14)(8,13)(9,15)(17,19,30,55,22,33,21,32,23,34,41,62,40,61,42,63)(18,24,44,64,28,47,26,29,27,46,39,54,43,52,50,53)(20,31,56,76,38,60,36,58,37,59,71,80,69,78,70,79)(25,45,68,77,49,73,48,51,35,65,72,75,67,74,66,57);(1,3,5,11,4,10,12,16)(2,7,8,9,6,14,13,15)(17,20,22,38,23,37,40,69)(18,25,43,67,27,35,28,49)(19,31,33,60,34,59,61,78)(21,36,41,71,42,70,30,56)(24,45,52,74,46,65,47,73)(26,48,44,68,50,66,39,72)(29,51,64,77,53,57,54,75)(32,58,62,80,63,79,55,76)

Polycyclic presentation:

2^9|a^2=f,b^2=dgh,c^2=eg,d^2=gi,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,f^c=fi

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, 42972);