#94237

order	512
nilpotency class	5
isoclinism	isoclinic class of #94192
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1391
commutator	C₂²×C₈	C₄²
Frattini, cosocle	#2931	C₂²
socle	C₂	#7237
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #2931 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

205 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 205 conjugacy classes of subgroups — too many to draw. Only its 48 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)(3,15,17,47)(4,10,21,38)(6,28,29,59)(7,14,27,41)(8,31,32,42)(9,33,34,18)(12,43,40,22)(13,26,45,30)(16,46,49,39)(19,35,53,61)(20,55,56,64)(23,44,25,58)(24,57,54,48)(36,60,63,50)(37,52,62,51);(1,3,6,18,30,54,22,50,7,19,23,51,8,20,4,16)(2,9,12,35,31,46,28,48,14,37,10,15,13,36,44,55)(5,24,25,49,26,53,21,33,27,56,29,60,32,17,43,52)(11,39,38,64,42,62,58,61,41,63,40,57,45,34,59,47)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fgi,e^2=fgh,g^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fh,g^a=gh,h^a=hi,d^b=dg,d^c=dh,e^d=ei,f^d=fi,g^d=gi

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, 1433);