#6512795

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499804
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3184 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)(15,23)(16,25)(17,21)(18,20)(19,22)(24,28)(26,27);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,26)(14,16,22,28)(17,25,18,24)(20,23,21,27);(6,9)(7,11)(13,16)(14,15)(17,23)(18,27)(19,28)(20,25)(21,24)(22,26);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,20,22,21)(15,24,26,25)(16,27,28,23);(6,9)(10,12)(13,17,19,18)(14,21,22,20)(15,24,26,25)(16,23,28,27)

Polycyclic presentation:

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

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, 6419991);