#111435

order	512
nilpotency class	3
isoclinism	isoclinic class of #10557
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

752 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,d^c=dhi,f^c=fi,g^c=gi

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, 18631);