#499709

order	512
nilpotency class	4
isoclinism	isoclinic class of #499263
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 432 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=e,d^2=efi,g^2=hi,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gh,h^b=hi,g^c=gh,h^c=hi,g^d=gi

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, 406905);