#99716

order	512
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#386
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

385 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=g,d^2=g,e^2=hi,f^2=ghi,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=ehi,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6912);