#6617407

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂×#3516	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2940 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)(13,14,19,16)(15,23,26,28)(17,21,18,22)(20,27,24,25);(6,10)(8,11)(13,15,19,26)(14,20,16,24)(17,25,18,27)(21,28,22,23);(2,4)(5,6)(7,8)(9,10)(11,12)(13,16,19,14)(15,24,26,20)(17,21,18,22)(23,25,28,27);(5,7,9,12)(6,11,10,8)(13,17,19,18)(14,21,16,22)(15,25,26,27)(20,28,24,23);(5,8,9,11)(6,7,10,12)(13,17,19,18)(14,22,16,21)(15,25,26,27)(20,23,24,28)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=hi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,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, 6524603);