#93704

order	512
nilpotency class	5
isoclinism	isoclinic class of #93701
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1316
commutator	C₂³×C₄	C₂×C₈
Frattini, cosocle	#1817	C₂²
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1817 ◁ G
LECS	G ▷ #1817 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 232 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=hi,d^2=f,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=chi,d^b=dehi,f^b=fi,d^c=dg,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 900);