#6467875

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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

2003 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:

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

Polycyclic presentation:

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

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, 6375071);