#233435

order	512
nilpotency class	3
isoclinism	isoclinic class of #233431
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
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

1622 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1622 conjugacy classes of subgroups — too many to draw. Only its 274 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140631);