#122593

order	512
nilpotency class	4
isoclinism	isoclinic class of #122544
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

293 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=hi,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^d=ehi

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, 29789);