#93553

order	512
nilpotency class	4
isoclinism	isoclinic class of #93553
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1300
commutator	C₂×C₄²	C₄²
Frattini, cosocle	C₂×#405	C₂²
socle	C₂	#7036
prime radical	C₂⁴	#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#405 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

254 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 56 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=f,f^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgh,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,h^b=hi,d^c=di,e^c=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, 749);