#93836

order	512
nilpotency class	5
isoclinism	isoclinic class of #93776
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1343
commutator	C₂×C₄×C₈	C₂×C₄
Frattini, cosocle	#1984	C₂²
prime radical	C₂³	#327
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1984 ◁ G
LECS	G ▷ #1984 ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁

168 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 30 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 — 2 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=h,c^2=fhi,d^2=h,e^2=h,f^2=hi,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gi,c^b=cf,d^b=defh,e^b=eh,f^b=fhi,d^c=dgh,e^d=ei

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1032);