#343033

order	512
nilpotency class	3
isoclinism	isoclinic class of #343027
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	#3497	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 747 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=efi,b^2=h,c^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=defh,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, 250229);