#263315

order	512
nilpotency class	3
isoclinism	isoclinic class of #263311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2397	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

906 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,d^c=dh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170511);