#475506

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1038 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1038 conjugacy classes of subgroups — too many to draw. Only its 234 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ei,c^2=h,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de,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, 382702);