#121162

order	512
nilpotency class	3
isoclinism	isoclinic class of #121161
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 338 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)(35,45,49,56)(36,50,54,60)(37,42)(38,43)(39,44)(40,47,57,48)(41,52,59,53)(46,62)(51,58)(55,63)(61,64);(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)(34,40)(36,51,54,64)(37,48)(38,47)(39,49)(41,58,59,61)(42,56)(43,45)(44,57)(46,53,63,52)(50,62,60,55);(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)(33,36,37,52,39,54,38,53)(34,41,42,50,44,59,43,60)(35,46,47,51,49,63,48,64)(40,55,45,61,57,62,56,58)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=hi,e^2=i,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=ei,g^b=gi,e^c=ehi,f^c=fh,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, 28358);