#110145

order	512
nilpotency class	3
isoclinism	isoclinic class of #9415
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

578 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 146 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)(3,7,10,15)(4,11,12,14)(5,9)(6,8)(13,16)(17,18,21,28,23,27,22,26)(19,29,32,38,33,39,31,24)(20,25,36,40,35,41,34,42)(30,43,44,37,45,47,46,48);(1,3)(2,7)(4,8)(5,10)(6,14)(9,15)(11,13)(12,16)(17,19,22,31,23,33,21,32)(18,24,26,39,27,38,28,29)(20,30,36,44,35,45,34,46)(25,37,40,47,41,48,42,43);(1,4,6,15,5,12,13,7)(2,3,8,14,9,10,16,11)(17,20,21,34,23,35,22,36)(18,25,26,40,27,41,28,42)(19,30,32,46,33,45,31,44)(24,37,39,47,38,48,29,43)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17341);