#94529

order	512
nilpotency class	6
isoclinism	isoclinic class of #94526
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7339
commutator	C₂×C₄²	C₂×C₈
Frattini, cosocle	#1822	C₂²
prime radical	C₂⁴	#100
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1822 ◁ G
LECS	G ▷ #1822 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

198 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 24 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=h,d^2=f,e^2=i,f^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ch,d^b=dehi,e^b=ei,f^b=fhi,d^c=dgi,f^c=fi,e^d=eh,g^d=gi

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1725);