#131768

order	512
nilpotency class	4
isoclinism	isoclinic class of #131664
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#62373	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

556 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 256

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=hi,e^2=gi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,f^b=fgh,h^b=hi,e^c=egi,f^c=fg

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38964);