#213935

order	512
nilpotency class	3
isoclinism	isoclinic class of #213876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

777 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 777 conjugacy classes of subgroups — too many to draw. Only its 186 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=hi,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,f^b=fh,g^b=gh,g^c=gi,g^d=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121131);