#125060

order	512
nilpotency class	4
isoclinism	isoclinic class of #124987
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1536
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#376	C₂³
socle	C₂	#7653
prime radical	C₂³	#410
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ #376 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

254 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,13,16,38,17,39,27,55)(4,18)(5,23,25,56)(6,11)(7,12)(8,15,29,44,31,24,30,45)(9,32)(10,26,35,49)(14,40,42,37,43,61,54,62)(19,36)(20,50,52,33)(21,34)(22,48)(28,57,46,41,59,63,58,51)(47,53,64,60);(1,3,7,17)(2,8)(4,14,22,43)(5,24,25,15)(6,27,19,16)(9,28)(10,13)(11,29)(12,31)(18,46)(20,51,52,41)(21,54,32,42)(23,55)(26,45,49,44)(30,36)(33,37)(34,59)(35,39)(38,56)(40,47)(48,58)(50,62)(53,57,60,63)(61,64);(1,4,19,32,7,22,6,21)(2,9,11,18,12,34,36,48)(3,14,16,42,17,43,27,54)(5,20,49,60,25,52,26,53)(8,28,29,46,31,59,30,58)(10,33,23,47,35,50,56,64)(13,37,55,40,39,62,38,61)(15,41,45,57,24,51,44,63)

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=fgh,d^2=gi,e^2=h,f^2=i,b^a=bd,c^a=cfgh,d^a=df,g^a=gi,d^b=dg,e^b=efg,f^b=fi,h^b=hi,e^d=ei

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, 32256);