#233765

order	512
nilpotency class	3
isoclinism	isoclinic class of #233754
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1292 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1292 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,c^b=cg,d^b=dhi,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=fhi

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, 140961);