#93123

order	512
nilpotency class	4
isoclinism	isoclinic class of #93119
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂³×C₄	C₄²
Frattini, cosocle	C₂²×#125	C₂²
prime radical	C₂⁴	#95
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#125 ◁ G
LECS	G ▷ C₂²×#125 ▷ C₂⁴ ▷ C₂² ▷ C₁

299 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 42 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)(3,11)(4,15,16,8)(5,9)(6,10)(7,12,19,25)(13,23)(14,24)(17,28,29,21)(18,22)(20,26,30,32)(27,31)(33,34,37,41,38,42,49,56)(35,43,45,57,46,53,59,63)(36,40,47,54,48,55,60,64)(39,50,51,58,52,61,62,44);(1,3,5,13,6,14,18,27)(2,7,9,20,10,19,22,30)(4,12,17,26,16,25,29,32)(8,11,21,23,15,24,28,31)(33,35)(34,39)(36,44)(37,45,49,59)(38,46)(40,53)(41,51,56,62)(42,52)(43,55)(47,50,60,61)(48,58)(54,63,64,57)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=i,e^2=h,b^a=bc,c^a=cf,e^a=egi,f^a=fh,g^a=gi,c^b=cg,d^b=dfi,f^b=fi,d^c=dh,e^d=ei

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, 319);