#343554

order	512
nilpotency class	3
isoclinism	isoclinic class of #333271
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 566 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)(3,4)(5,7)(6,8)(9,10,12,16,13,17,20,15)(11,14,18,22,19,23,24,21)(25,26)(27,35)(28,33)(29,31)(30,34)(32,37)(36,39)(38,40);(1,3)(2,6)(4,7)(5,8)(9,11,13,19)(10,14,17,23)(12,18,20,24)(15,21,16,22)(26,31)(27,36)(28,30)(32,39)(35,40)(37,38);(1,4,5,6)(2,3,7,8)(10,15)(12,20)(14,21)(16,17)(18,24)(22,23)(25,27)(26,32)(28,36)(29,37)(30,38)(31,35)(33,40)(34,39);(2,7)(3,8)(10,15)(12,20)(14,21)(16,17)(18,24)(22,23)(25,28,29,30)(26,33,31,34)(27,36,37,38)(32,40,35,39)

Polycyclic presentation:

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

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, 250750);