#329648

order	512
nilpotency class	3
isoclinism	isoclinic class of #329647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
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

717 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 717 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=dh,g^b=gi,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 236844);