#6598607

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#38032	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1864 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:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=f,d^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,e^b=ehi,d^c=dhi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6505803);