#135849

order	512
nilpotency class	4
isoclinism	isoclinic class of #134691
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

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

259 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 92 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)(5,6,11,15,10,16,34,47,12,18,36,48,33,17,9,31)(7,19,21,13,22,39,57,38,24,51,58,40,23,59,32,49)(8,25,29,62,28,46,27,43,30,45,35,61,66,14,41,44)(20,52,54,68,26,63,65,37,56,67,53,42,55,50,64,60);(5,7,11,21,10,22,34,57,12,24,36,58,33,23,9,32)(6,13,15,39,16,38,47,51,18,40,48,59,17,49,31,19)(8,26,29,65,28,56,27,53,30,55,35,64,66,20,41,54)(14,42,44,50,25,60,62,52,46,68,43,63,45,37,61,67);(2,4)(5,8,10,28,12,30,33,66)(6,14,17,45,18,46,16,25)(7,20,23,55,24,56,22,26)(9,27,11,35,34,41,36,29)(13,37,38,42,40,60,49,68)(15,43,31,62,48,44,47,61)(19,50,39,52,51,63,59,67)(21,53,32,65,58,54,57,64)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43045);