#244373

order	512
nilpotency class	3
isoclinism	isoclinic class of #233469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

754 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=hi,c^2=hi,d^2=efhi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=dh,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi

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, 151569);