#512466

order	512
nilpotency class	4
isoclinism	isoclinic class of #511954
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2250	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

729 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

(2,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,17)(18,27)(19,23)(20,28)(22,25)(24,26);(6,9)(10,12)(13,14,19,16,20,25,18,24,21,15,26,27,22,28,17,23);(1,2,3,4)(13,15,26,16,20,28,17,24,21,14,19,27,22,25,18,23);(5,7,8,11)(6,10,9,12)(13,16,17,14,22,23,26,28,21,27,18,15,20,24,19,25)

Polycyclic presentation:

2^9|b^2=g,c^2=fh,d^2=e,e^2=hi,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,g^a=ghi,h^a=hi,c^b=ci,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419662);