#6449107

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437995
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	#62815	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3432 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)(9,10,14,18)(11,15,21,29)(12,16,24,31)(13,25,26,17)(19,27,22,35)(20,33,34,28)(23,37,38,30)(32,40,36,39);(9,11)(10,15)(12,22,24,19)(13,20)(14,21)(16,27,31,35)(17,28)(18,29)(23,36,38,32)(25,33)(26,34)(30,39,37,40);(2,5)(6,8)(9,12)(10,16)(11,19)(13,23)(14,24)(15,27)(17,30)(18,31)(20,32)(21,22)(25,37)(26,38)(28,39)(29,35)(33,40)(34,36);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,23)(14,24)(15,27)(17,30)(18,31)(20,32)(21,22)(25,37)(26,38)(28,39)(29,35)(33,40)(34,36);(9,13,14,26)(10,17,18,25)(11,20,21,34)(12,23,24,38)(15,28,29,33)(16,30,31,37)(19,32,22,36)(27,39,35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284505, #72286090, #72286206
maximal normal subgroup of	#72284505, #72286090, #72286206

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356303);