#129309

order	512
nilpotency class	4
isoclinism	isoclinic class of #12727
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1782
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8300
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

621 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 621 conjugacy classes of subgroups — too many to draw. Only its 116 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,f^b=fgi,e^c=ei,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, 36505);