#6440463

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437621
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	#62811	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4332 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306425, #72306427, #72306431
maximal normal subgroup of	#72306425, #72306427, #72306431

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347659);