#93548

order	512
nilpotency class	4
isoclinism	isoclinic class of #93529
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1284
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂×#405	C₂²
socle	C₂	#7036
prime radical	C₂⁴	#104
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=f,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efghi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,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, 744);