#454923

order	512
nilpotency class	3
isoclinism	isoclinic class of #454045
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1804 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=e,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,h^c=hi,f^d=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, 362119);