#145519

order	512
nilpotency class	4
isoclinism	isoclinic class of #145511
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#469
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

271 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,10)(4,12)(5,13)(6,9)(7,11)(8,15)(14,16)(17,18,25,32)(19,27,37,47)(20,38,33,26)(21,29,41,45)(22,28,39,48)(23,31,24,30)(34,43,40,44)(35,42,36,46);(1,3)(2,7)(4,8)(5,14)(6,12)(9,16)(10,13)(11,15)(17,19,25,37)(18,26,32,38)(20,22,33,39)(21,40,41,34)(23,36,24,35)(27,45,47,29)(28,46,48,42)(30,44,31,43);(1,4,5,7,6,10,15,16)(2,3,8,11,9,12,13,14)(17,20,39,19,23,34,21,35,25,33,22,37,24,40,41,36)(18,27,45,38,30,46,28,44,32,47,29,26,31,42,48,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=dh,d^2=gi,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fhi,g^b=gi,h^b=hi,e^c=eghi,f^c=fgh

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, 52715);