#6617575

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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

3314 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)(13,14,19,27,18,23,16,21)(15,22,25,20,17,26,24,28);(6,10)(8,11)(13,15,18,17)(14,20,23,28)(16,25,19,24)(21,26,27,22);(2,4)(5,6)(7,8)(9,10)(11,12)(13,16,18,19)(14,21,23,27)(15,24,17,25)(20,22,28,26);(5,7,9,12)(6,11,10,8)(13,17,18,15)(14,22,23,26)(16,25,19,24)(20,21,28,27);(5,8,9,11)(6,7,10,12)(15,17)(20,28)(22,26)(24,25)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6524771);