#6606076

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2124 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)(15,20)(16,25)(17,21)(18,27)(19,22)(23,28)(24,26);(2,4)(5,6,8,9)(7,10,11,12)(13,15,19,18)(14,20,22,27)(16,21,26,23)(17,24,28,25);(6,9)(7,11)(13,16,15,23,19,26,18,21)(14,17,27,24,22,28,20,25);(5,7)(6,10)(8,11)(9,12)(13,17,19,28)(14,21,22,23)(15,24,18,25)(16,20,26,27);(6,9)(7,11)(13,15,19,18)(14,20,22,27)(16,23,26,21)(17,25,28,24)

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,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, 6513272);