#110142

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

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

normal subgroups only

This group has 598 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)(19,28)(20,25,33,41)(21,35)(22,26)(23,27)(24,30)(29,42,44,48)(31,39)(32,45,34,40)(36,38)(37,46,47,43);(1,3)(2,7)(4,8)(5,10)(6,14)(9,15)(11,13)(12,16)(17,19,22,30,23,31,21,36)(18,24,26,39,27,38,35,28)(20,29,34,43,33,44,32,46)(25,37,40,48,41,47,45,42);(1,4,6,15,5,12,13,7)(2,3,8,14,9,10,16,11)(17,20)(18,25,27,41)(19,29)(21,32)(22,34)(23,33)(24,37,38,47)(26,40,35,45)(28,42,39,48)(30,43)(31,44)(36,46)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=i,e^2=hi,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, 17338);