#93166

order	512
nilpotency class	4
isoclinism	isoclinic class of #93165
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²×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	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×#126 ▷ C₂⁴ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 340 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)(3,14,16,32)(4,19,20,35)(5,23,25,10)(6,13)(7,12)(8,15,28,41)(9,22,33,51)(11,27)(17,39)(18,40,46,56)(21,52,53,36)(24,49,50,34)(26,37,55,38)(29,47,59,61)(30,43,31,44)(42,57,62,54)(45,58)(48,60)(63,64);(1,3,5,17,7,18,26,48)(2,8,10,29,12,30,38,59)(4,15,21,42,20,43,50,62)(6,16,25,45,27,46,55,63)(9,32,34,58,33,56,52,64)(11,28,37,57,13,31,23,54)(14,36,39,19,40,49,60,35)(22,44,24,47,51,41,53,61)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,e^2=h,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, 362);