#147852

order	512
nilpotency class	4
isoclinism	isoclinic class of #147850
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10995
commutator	C₂×#41	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#60753	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1069 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,e^b=ehi,f^b=fi,g^b=gh,d^c=dh,f^c=fi,e^d=ei,g^d=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55048);