#238886

order	512
nilpotency class	3
isoclinism	isoclinic class of #238875
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

835 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 835 conjugacy classes of subgroups — too many to draw. Only its 180 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 — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146082);