#6591385

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573549
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
prime radical	C₂²×#547	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4341 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)(5,6,8,10)(7,9,11,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,24)(25,27)(26,28);(5,7)(6,9)(8,11)(10,12)(13,15,18,25)(14,19,22,28)(16,24,17,26)(20,27,21,23);(2,4)(7,11)(9,12)(13,16,18,17)(14,20,22,21)(15,24,25,26)(19,27,28,23);(5,7)(6,9)(8,11)(10,12)(13,16,18,17)(14,21,22,20)(15,24,25,26)(19,23,28,27);(7,11)(9,12)(13,16,18,17)(14,20,22,21)(15,24,25,26)(19,27,28,23)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6498581);