#343052

order	512
nilpotency class	3
isoclinism	isoclinic class of #343051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2679
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3496	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

767 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 767 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,f^d=fi

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, 250248);