#110110

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#120
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

596 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=i,e^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi,e^c=eh,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17306);