#6687245

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682500
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2186 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei

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, 6594441);