#147848

order	512
nilpotency class	4
isoclinism	isoclinic class of #147846
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

1067 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:

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

Polycyclic presentation:

2^9|a^2=g,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=eh,f^b=fi,g^b=gh,d^c=dh,f^c=fi,g^c=gi,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, 55044);