#128890

order	512
nilpotency class	4
isoclinism	isoclinic class of #11915
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1533
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×#97	C₂³
socle	C₂	#8224
prime radical	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

285 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 90 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)(4,10,21,36,24,38,23,56)(6,26,25,40,28,41,29,12)(15,42,44,31,20,52,46,35)(17,48,47,43,51,33,50,57)(22,54,53,39,27,58,55,37)(45,60,63,61,49,59,64,62);(1,3,6,17,7,19,29,50,8,18,28,51,5,16,25,47)(2,9,12,33,11,32,26,57,13,34,40,48,30,14,41,43)(4,20,22,49,23,44,55,63,24,15,27,45,21,46,53,64)(10,35,37,60,36,42,54,61,38,31,39,59,56,52,58,62);(1,4,5,21,8,24,7,23)(2,10,11,36,13,38,30,56)(3,15,16,44,18,20,19,46)(6,27,25,55,28,22,29,53)(9,31,32,52,34,35,14,42)(12,39,26,58,40,37,41,54)(17,49,47,64,51,45,50,63)(33,60,57,61,48,59,43,62)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,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=cg,d^b=dh,g^b=gi,f^c=fi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36086);