#121132

order	512
nilpotency class	3
isoclinism	isoclinic class of #121129
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

327 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 327 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=ei,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, 28328);