#6506097

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500186
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×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1449 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)(3,8)(4,6)(5,7)(17,18,24,32,23,26,22,25)(19,27,28,31,21,20,29,30);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,20)(21,23)(22,29)(24,28)(25,31)(26,27)(30,32);(9,11)(10,13)(12,16)(14,15)(17,20,23,27)(18,19,26,21)(22,30,24,31)(25,28,32,29);(11,16)(13,15)(17,21,23,19)(18,20,26,27)(22,28,24,29)(25,31,32,30);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,22,23,24)(18,25,26,32)(19,28,21,29)(20,30,27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6413293);