#134681

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

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,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, 41877);