#505073

order	512
nilpotency class	4
isoclinism	isoclinic class of #504384
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1771 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

409 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=e,d^2=efhi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412269);