#290939

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

717 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 717 conjugacy classes of subgroups — too many to draw. Only its 136 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,23,27)(19,24,30,41)(20,31,33,44)(21,35,22,26)(25,37,43,34)(28,42,45,48)(29,40,36,39)(32,47,46,38);(1,3,4,8)(2,5,6,7)(11,16)(12,15)(17,19)(18,24)(20,32)(21,29)(22,36)(23,30)(25,42)(26,39)(27,41)(28,34)(31,47)(33,46)(35,40)(37,45)(38,44)(43,48);(9,11,15,10,13,16,12,14)(17,20,22,37,23,33,21,34)(18,25,35,44,27,43,26,31)(19,28,36,32,30,45,29,46)(24,38,40,48,41,47,39,42);(2,6)(5,7)(20,33)(25,43)(28,45)(31,44)(32,46)(34,37)(38,47)(42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198135);