#127693

order	512
nilpotency class	4
isoclinism	isoclinic class of #11912
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1530
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×#97	C₂³
socle	C₂	#8034
prime radical	C₂⁴	#101
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

372 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34889);