#6454908

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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 ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284872, #72285147
maximal normal subgroup of	#72284872, #72285147

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6362104);