#298581

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1146 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:

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

Polycyclic presentation:

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

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, 205777);