#233816

order	512
nilpotency class	3
isoclinism	isoclinic class of #233668
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

1278 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1278 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)(18,32)(19,23)(20,27)(25,40)(26,34)(28,37)(29,35)(30,38)(31,36)(33,42)(39,43)(41,44);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,15,19,28)(14,20,23,34)(16,25,31,43)(17,27,18,26)(21,33,36,44)(22,37,32,24)(29,42,30,41)(35,39,38,40);(2,4)(5,7,10,6)(8,12,9,11)(13,16,19,31)(14,21,23,36)(15,25,28,43)(17,30,18,29)(20,33,34,44)(22,38,32,35)(24,39,37,40)(26,42,27,41);(13,17,19,18)(14,22,23,32)(15,26,28,27)(16,29,31,30)(20,24,34,37)(21,35,36,38)(25,41,43,42)(33,40,44,39)

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=di,e^b=eh,f^b=fi,d^c=di,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, 141012);