#93164

order	512
nilpotency class	4
isoclinism	isoclinic class of #93155
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

323 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 44 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 360);