#134682

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,d^2=gi,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^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, 41878);