#93131

order	512
nilpotency class	4
isoclinism	isoclinic class of #93129
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂⁴

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	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×#126 ▷ C₂⁴ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 332 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)(3,14,16,35)(4,20,22,10)(5,26,27,11)(6,30)(7,12)(8,13)(9,15)(17,48,49,61)(18,42,19,41)(21,32)(23,54,43,36)(24,37,25,51)(28,38,29,58)(31,44,56,63)(33,46)(34,45)(39,53,57,55)(40,47,62,50)(52,60,64,59);(1,3,5,17,7,16,29,47)(2,9,11,31,12,33,38,60)(4,21,23,52,22,45,53,63)(6,19,28,49,8,18,27,50)(10,35,36,61,20,14,39,40)(13,34,26,56,30,32,58,59)(15,43,44,25,46,55,64,24)(37,41,57,62,51,42,54,48)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=hi,e^2=h,b^a=bc,c^a=cf,e^a=eghi,f^a=fh,g^a=gi,c^b=cg,d^b=dfhi,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, 327);