#146188

order	512
nilpotency class	4
isoclinism	isoclinic class of #146186
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2092
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	Q₈
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 213 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) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=dei,d^2=gi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,d^b=dg,e^b=egi,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53384);