#93134

order	512
nilpotency class	4
isoclinism	isoclinic class of #93129
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×#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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=hi,d^2=i,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, 330);