#121209

order	512
nilpotency class	3
isoclinism	isoclinic class of #121207
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1443
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

334 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 108 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)(4,9)(5,20)(6,11)(7,10)(8,14)(13,28)(15,29)(16,23)(17,22)(18,26)(19,27)(21,25)(24,30)(31,32)(33,34,37,42,38,41,50,57)(35,39,45,53,44,52,49,58)(36,46,48,61,47,40,54,56)(43,51,60,63,59,55,62,64);(1,3,7,16,6,15,5,14)(2,8,20,29,11,23,10,12)(4,17,18,30,19,31,21,13)(9,24,25,22,27,28,26,32)(33,35)(34,39,41,52)(36,43,47,59)(37,49)(38,44)(40,55)(42,58,57,53)(45,50)(46,51)(48,62,54,60)(56,63)(61,64);(1,4,5,18,6,19,7,21)(2,9,10,25,11,27,20,26)(3,13,14,17,15,30,16,31)(8,22,12,28,23,32,29,24)(33,36)(34,40,41,46)(35,43,44,59)(37,48)(38,47)(39,51)(42,56,57,61)(45,60,49,62)(50,54)(52,55)(53,63)(58,64)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28405);