#93172

order	512
nilpotency class	4
isoclinism	isoclinic class of #93165
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:

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

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,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, 368);