#270817

order	512
nilpotency class	3
isoclinism	isoclinic class of #270811
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62111	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

929 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=i,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,g^b=gi,d^c=di,f^c=fh,g^c=gi,e^d=eh

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, 178013);