#6467109

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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 ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2196 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)(7,10)(8,9)(11,12)(13,14)(15,22)(16,21)(17,23)(18,27)(19,25)(20,24)(26,28);(2,4)(13,15,19,16)(14,21,25,22)(17,20,26,18)(23,27,28,24);(6,9)(10,12)(13,16,20,26,19,15,18,17)(14,22,24,23,25,21,27,28);(13,17,20,16,19,26,18,15)(14,23,24,21,25,28,27,22);(5,7,8,11)(6,10,9,12)(13,18,19,20)(14,24,25,27)(15,17,16,26)(21,23,22,28)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=h,e^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374305);