#6587780

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1625 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,22,23)(19,24,21,26)(20,28,30,25)(27,29,31,32);(2,6)(7,8)(17,19,22,21)(18,24,23,26)(20,29,30,32)(25,27,28,31);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,30)(18,25,23,28)(19,27,21,31)(24,32,26,29);(9,11,12,15)(10,14,13,16)(17,21,22,19)(18,26,23,24)(20,29,30,32)(25,27,28,31);(1,4,5,3)(2,8,6,7)(18,23)(19,26)(21,24)(25,28)(27,29)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285188, #72285214
maximal normal subgroup of	#72285188, #72285214

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6494976);