#357867

order	512
nilpotency class	3
isoclinism	isoclinic class of #184767
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×Q₈ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2594 conjugacy classes of subgroups; only the 570 normal subgroups are shown.

normal subgroups only

This group has 2594 conjugacy classes of subgroups — too many to draw. Only its 570 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,d^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^b=di,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265063);