#508856

order	512
nilpotency class	4
isoclinism	isoclinic class of #508669
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	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

464 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 118 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)(3,7)(4,5)(6,8)(9,10,14,20,16,22,13,18)(11,23,28,17,29,36,26,37)(12,30,15,33,32,19,31,21)(24,34,25,39,40,35,27,38);(2,5)(6,7)(9,11)(10,17)(12,25)(13,28)(14,26)(15,24)(16,29)(18,36)(19,34)(20,23)(21,38)(22,37)(27,32)(30,35)(31,40)(33,39);(1,3,4,8)(2,6,5,7)(10,18)(11,24)(12,31)(13,14)(15,32)(17,34)(19,30)(20,22)(23,39)(25,26)(27,28)(29,40)(35,37)(36,38);(3,8)(6,7)(9,12,13,31,16,32,14,15)(10,19,18,33,22,30,20,21)(11,25,28,40,29,27,26,24)(17,34,36,39,37,35,23,38)

Polycyclic presentation:

2^9|a^2=h,c^2=eh,d^2=efhi,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,e^b=ei,g^b=gh,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gh,h^c=hi,g^d=gi

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, 416052);