#6451747

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437445
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

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

3942 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)(3,8)(4,6)(5,7)(9,10)(11,19)(12,16)(13,25)(14,18)(15,22)(17,28)(20,36)(21,34)(23,35)(24,33)(26,30)(27,32)(29,39)(31,38)(37,40);(9,11)(10,15)(12,20)(13,26)(14,21)(16,31)(17,34)(18,30)(19,25)(22,28)(23,39)(24,37)(27,38)(29,33)(32,40)(35,36);(1,3,5,4)(2,6,7,8)(9,12,14,24)(10,16,18,33)(11,20,21,37)(13,27,28,23)(15,29,30,31)(17,35,25,32)(19,36,34,40)(22,39,26,38);(1,4,5,3)(2,6,7,8)(13,28)(15,30)(17,25)(22,26)(23,27)(29,31)(32,35)(38,39);(11,21)(15,30)(19,34)(20,37)(22,26)(29,31)(36,40)(38,39)

Polycyclic presentation:

2^9|c^2=gi,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,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	#72257430
maximal normal subgroup of	#72257430
maximal quotient of	#72264529

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6358943);