#94444

order	512
nilpotency class	5
isoclinism	isoclinic class of #94441
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7307
commutator	C₂×C₄²	C₄²
Frattini, cosocle	C₂×#536	C₂²
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂×#536 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

433 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 433 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=hi,d^2=i,e^2=i,f^2=i,b^a=bc,c^a=cf,e^a=eghi,f^a=fh,h^a=hi,c^b=cg,d^b=dfh,g^b=gi,d^c=dhi,e^c=ei,e^d=ei,f^d=fi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1640);