#318363

order	512
nilpotency class	3
isoclinism	isoclinic class of #262851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

658 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=eghi,c^2=i,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 225559);