#319476

order	512
nilpotency class	3
isoclinism	isoclinic class of #298795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

780 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 182 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)(3,7)(4,6)(5,8)(9,10,12,15,13,16,21,24)(11,17,18,22,19,23,20,14)(25,26)(27,31)(28,37)(29,32)(30,33)(34,38)(35,39)(36,40);(1,3,4,8)(2,5,6,7)(25,27,34,35,30,28,29,36)(26,31,38,39,33,37,32,40);(9,11,13,19)(10,14,16,22)(12,20,21,18)(15,23,24,17)(25,28,30,27)(26,31,33,37)(29,36,34,35)(32,39,38,40);(2,6)(5,7)(25,29,30,34)(26,32,33,38)(27,35,28,36)(31,39,37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 226672);