#6453840

order	512
nilpotency class	3
isoclinism	isoclinic class of #23912
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	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3099 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)(11,21)(12,17)(13,18)(14,28)(15,20)(16,23)(19,30)(22,37)(24,36)(25,35)(26,40)(27,34)(29,32)(31,39)(33,38);(9,11)(10,16)(12,22)(13,26)(14,29)(15,24)(17,33)(18,31)(19,36)(20,32)(21,28)(23,30)(25,39)(27,38)(34,40)(35,37);(2,5)(6,8)(9,12,15,13)(10,17,20,18)(11,22,24,26)(14,27,30,25)(16,31,32,33)(19,35,28,34)(21,37,36,40)(23,39,29,38);(9,13,15,12)(10,18,20,17)(11,22,24,26)(14,25,30,27)(16,31,32,33)(19,34,28,35)(21,37,36,40)(23,39,29,38);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317850
maximal normal subgroup of	#72317850

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361036);