#241634

order	512
nilpotency class	3
isoclinism	isoclinic class of #233754
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

761 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 761 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 256

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dh,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148830);