#146272

order	512
nilpotency class	4
isoclinism	isoclinic class of #146230
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2094
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

309 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(9,10,14,35,15,22,13,20,17,24,33,21,16,23,34,39)(11,25,32,38,28,40,26,19,12,30,27,18,29,36,31,37);(1,2,4,7)(3,5,8,6)(9,11,15,29,17,12,16,28)(10,18,23,37,24,38,22,19)(13,32,33,31,34,27,14,26)(20,36,35,25,39,40,21,30);(2,5)(3,8)(6,7)(9,12)(10,19)(11,17)(13,27)(14,31)(15,28)(16,29)(18,23)(20,30)(21,40)(22,38)(24,37)(25,39)(26,33)(32,34)(35,36)

Polycyclic presentation:

2^9|a^2=dg,b^2=deghi,d^2=g,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=dg,e^c=eg,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224921, #72224926, #72224927
maximal normal subgroup of	#72224921, #72224926, #72224927

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53468);