#111281

order	512
nilpotency class	3
isoclinism	isoclinic class of #10026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ 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:

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

Polycyclic presentation:

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

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, 18477);