#110101

order	512
nilpotency class	3
isoclinism	isoclinic class of #110086
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 659 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=eh,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, 17297);