#146212

order	512
nilpotency class	4
isoclinism	isoclinic class of #143142
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#493
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2093
prime radical	C₂³	#480
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

229 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 229 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53408);