#94546

order	512
nilpotency class	6
isoclinism	isoclinic class of #7523
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1404
commutator	#125	C₂×C₈
Frattini, cosocle	#1995	C₂²
socle	C₂	#7345
prime radical	#3531	C₄
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1995 ◁ G
LECS	G ▷ #1995 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

199 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 199 conjugacy classes of subgroups — too many to draw. Only its 26 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:

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

Polycyclic presentation:

2^9|a^2=d,c^2=h,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ch,d^b=dei,e^b=ehi,g^b=gh,d^c=dgh,e^c=eh,g^c=gi,e^d=ehi,g^d=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1742);