#318478

order	512
nilpotency class	3
isoclinism	isoclinic class of #262915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 225674);