#245420

order	512
nilpotency class	3
isoclinism	isoclinic class of #245375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1228 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)(3,6,4,7)(9,10)(11,20)(12,26)(13,29)(14,18)(15,19)(16,23)(17,27)(21,25)(22,31)(24,32)(28,33)(30,34)(35,37)(36,39)(38,40);(1,3,5,4)(2,6,8,7)(9,11,15,25)(10,16,19,32)(12,22,28,38)(13,24,14,23)(17,31,34,40)(18,21,29,20)(26,35,33,36)(27,39,30,37);(2,7)(3,4)(6,8)(9,10)(11,21)(12,17)(13,18)(14,29)(15,19)(16,23)(20,25)(22,36)(24,32)(26,27)(28,34)(30,33)(31,39)(35,38)(37,40);(9,12,15,28)(10,17,19,34)(11,22,25,38)(13,30,14,27)(16,31,32,40)(18,33,29,26)(20,35,21,36)(23,39,24,37)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152616);