#6572256

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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

2999 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)(14,19)(15,17)(20,22)(21,26)(23,24)(27,28);(5,6,10,9)(7,12,8,11)(13,14)(15,19)(16,21)(17,20)(18,22)(23,26)(24,27)(25,28);(2,4)(5,7,10,8)(6,11,9,12)(13,15,18,17)(14,20,22,19)(16,24,25,23)(21,26,28,27);(5,8,10,7)(6,11,9,12)(13,16,18,25)(14,21,22,28)(15,23,17,24)(19,26,20,27);(5,9,10,6)(7,12,8,11)(13,15,18,17)(14,19,22,20)(16,24,25,23)(21,27,28,26)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6479452);