#6454905

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

3278 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,e^2=g,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	#72285079
maximal normal subgroup of	#72285079

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6362101);