#93128

order	512
nilpotency class	4
isoclinism	isoclinic class of #93119
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂³×C₄	C₄²
Frattini, cosocle	C₂²×#126	C₂²
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×#126 ▷ C₂⁴ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 315 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=h,e^2=hi,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, 324);