#6441585

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1862 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)(11,13)(12,14)(15,16)(17,18,22,25)(19,26,29,20)(21,31,23,24)(27,32,28,30);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(19,27)(20,30)(21,23)(24,31)(26,32)(28,29);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,19)(18,20)(21,27)(22,29)(23,28)(24,32)(25,26)(30,31);(1,4,5,3)(2,8,6,7)(17,20)(18,19)(21,30)(22,26)(23,32)(24,28)(25,29)(27,31);(17,21,22,23)(18,24,25,31)(19,28,29,27)(20,30,26,32)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,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,e^b=ei,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, 6348781);