#135016

order	512
nilpotency class	4
isoclinism	isoclinic class of #134965
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1808
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#466
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

248 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 98 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)(10,16)(14,15)(17,18)(19,29)(20,35,37,25)(21,39)(22,42)(23,27)(24,32)(26,30)(28,40)(31,56,58,49)(33,62)(34,51)(36,66,68,69)(38,47,67,65)(41,70)(43,57,63,79)(44,61)(45,72)(46,54,64,76)(48,55)(50,53)(52,60)(59,74,78,77)(71,80,73,75);(2,6)(7,15)(8,13)(9,16)(17,19,30,53,22,33,41,61,23,34,55,72,40,60,21,32)(18,24,39,52,28,45,48,51,27,44,70,62,42,50,26,29)(20,31,54,75,38,59,68,79,37,58,76,80,67,78,36,57)(25,43,69,74,47,71,64,56,35,63,66,77,65,73,46,49);(1,3,4,10)(2,7,6,15)(5,14,12,11)(8,16,13,9)(17,20,23,37)(18,25,27,35)(19,31,34,58)(21,36,41,68)(22,38,40,67)(24,43,44,63)(26,46,48,64)(28,47,42,65)(29,49,51,56)(30,54,55,76)(32,57,61,79)(33,59,60,78)(39,69,70,66)(45,71,50,73)(52,74,62,77)(53,75,72,80)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42212);