#6454608

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3121 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361804);