#138093

order	512
nilpotency class	4
isoclinism	isoclinic class of #138092
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9124
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2875 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

314 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=g,d^2=ghi,e^2=i,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,h^a=hi,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,d^c=di,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45289);