#134665

order	512
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1568
prime radical	C₂⁴	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

324 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41861);