#93582

order	512
nilpotency class	4
isoclinism	isoclinic class of #93565
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1286
commutator	C₂×C₄²	C₄²
Frattini, cosocle	C₂×#405	C₂²
socle	C₂	#7039
prime radical	C₂⁴	#101
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#405 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

262 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=fg,e^2=i,f^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgh,h^a=hi,c^b=cg,d^b=dh,g^b=gi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 778);