#121861

order	512
nilpotency class	3
isoclinism	isoclinic class of #121861
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1445
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 365 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)(3,12)(4,17,21,27)(5,22)(6,10)(7,11)(8,14)(9,20,28,19)(13,29,31,23)(15,25)(16,24)(18,32,30,26)(33,34)(35,39,44,54)(36,46,48,61)(37,42)(38,41)(40,47,56,63)(43,58)(45,53,49,57)(50,51)(52,62)(55,60)(59,64);(1,3)(2,8,11,25)(4,18)(5,14)(6,15)(7,16)(9,26,28,32)(10,12,22,24)(13,19)(17,29,27,23)(20,31)(21,30)(33,35,37,49,38,44,50,45)(34,39,51,53,41,54,42,57)(36,43,47,62,48,59,63,60)(40,55,61,64,56,52,46,58);(1,4)(2,9,11,28)(3,13)(5,19)(6,20)(7,21)(8,23,25,29)(10,17,22,27)(12,26,24,32)(14,30)(15,18)(16,31)(33,36,37,47,38,48,50,63)(34,40,51,61,41,56,42,46)(35,43,49,62,44,59,45,60)(39,52,53,58,54,55,57,64)

Polycyclic presentation:

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

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, 29057);