#213933

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, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 753 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) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,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, 121129);