#120889

order	512
nilpotency class	3
isoclinism	isoclinic class of #120886
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

371 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 114 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)(3,13,15,28,16,8,22,24)(4,17,20,27,19,9,21,26)(14,23,31,32,30,29,18,25)(33,34)(35,39,44,51)(36,40,47,54)(37,42)(38,41)(43,50)(45,52,58,63)(46,53)(48,55,60,61)(49,56)(57,62)(59,64);(1,3)(2,8)(4,18,19,31)(5,15)(6,22)(7,16)(9,25,17,32)(10,13)(11,24)(12,28)(14,20,30,21)(23,26,29,27)(33,35)(34,39,41,51)(36,46,47,59)(37,45)(38,44)(40,53)(42,52,56,63)(43,48,57,60)(49,58)(50,61)(54,64)(55,62);(1,4)(2,9,10,17)(3,14)(5,21)(6,20)(7,19)(8,23,13,29)(11,27,12,26)(15,31)(16,30)(18,22)(24,32,28,25)(33,36)(34,40,41,54)(35,43,44,57)(37,48)(38,47)(39,50)(42,55,56,61)(45,46,58,59)(49,60)(51,62)(52,53)(63,64)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,g^b=gh,e^c=eh,f^c=fhi,g^c=gi

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, 28085);