#132075

order	512
nilpotency class	4
isoclinism	isoclinic class of #132074
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8459
commutator	#117	C₂²×C₄
Frattini, cosocle	#405	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ #117 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

486 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=i,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,d^b=dh,e^b=ehi,f^b=fghi,g^b=gi,d^c=dh,e^d=ei,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39271);