#128442

order	512
nilpotency class	3
isoclinism	isoclinic class of #106017
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1749
prime radical	C₂×#41	C₄²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

276 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 256

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,e^2=g,f^2=h,h^2=i,b^a=bd,d^a=di,f^a=fg,c^b=ch,d^b=dg,e^b=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35638);