#6542714

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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

3243 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6449910);