#499689

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

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

510 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 130 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=efh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=dh,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, 406885);