#6441876

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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
prime radical	#35564	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2039 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,25)(19,26,29,20)(21,31,23,24)(27,30,32,28);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,18)(19,27)(20,30)(21,24)(22,25)(23,31)(26,28)(29,32);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,19,22,29)(18,20,25,26)(21,30,23,28)(24,32,31,27);(1,4,5,3)(2,6,8,7)(17,20,22,26)(18,19,25,29)(21,27,23,32)(24,28,31,30);(17,21,22,23)(18,24,25,31)(19,28,29,30)(20,27,26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349072);