#6524412

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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	#28237	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1511 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei,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, 6431608);