#6598551

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	#38033	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1888 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)(13,14,20,17)(15,25,28,16)(18,23,19,24)(21,27,26,22);(5,6,8,9)(7,10,11,12)(13,15,19,27,20,28,18,22)(14,21,23,16,17,26,24,25);(2,4)(6,9)(7,11)(13,16,18,21,20,25,19,26)(14,22,24,28,17,27,23,15);(5,7)(6,10)(8,11)(9,12)(13,17,20,14)(15,26,28,21)(16,27,25,22)(18,23,19,24);(6,9)(7,11)(13,18,20,19)(14,23,17,24)(15,27,28,22)(16,21,25,26)

Polycyclic presentation:

2^9|a^2=i,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,e^b=eh,d^c=dhi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6505747);