#6454046

order	512
nilpotency class	3
isoclinism	isoclinic class of #23903
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
socle	C₂²	C₂²×#120
prime radical	#62781	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,26)(20,27)(21,28)(22,30)(23,31)(24,37)(25,32)(29,34)(33,38)(35,39)(36,40);(9,11,14,21)(10,15,18,28)(12,19,23,34)(13,24,25,20)(16,29,31,26)(17,27,32,37)(22,35,36,33)(30,39,40,38);(2,5)(6,8)(9,12,14,23)(10,16,18,31)(11,19,21,34)(13,22,25,36)(15,26,28,29)(17,30,32,40)(20,33,24,35)(27,38,37,39);(9,13)(10,17)(11,20)(12,22)(14,25)(15,27)(16,30)(18,32)(19,33)(21,24)(23,36)(26,38)(28,37)(29,39)(31,40)(34,35);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=g,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dhi,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257536, #72266610
maximal normal subgroup of	#72257536, #72266610
maximal quotient of	#72264504

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361242);