#121174

order	512
nilpotency class	3
isoclinism	isoclinic class of #121173
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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

350 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 104 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:

(1,2,7,10,6,11,5,21)(3,12,16,23,15,28,14,8)(4,9,19,25,20,26,18,32)(13,27,30,24,17,31,29,22)(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);(1,3,7,16,6,15,5,14)(2,8,21,28,11,23,10,12)(4,17,18,30,20,13,19,29)(9,24,25,31,26,22,32,27)(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);(1,4,5,18,6,20,7,19)(2,9,10,25,11,26,21,32)(3,13,14,29,15,17,16,30)(8,22,12,27,23,24,28,31)(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=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=gi,e^c=ehi,g^c=gi

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, 28370);