#6451884

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437632
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	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3996 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)(11,16)(12,17)(13,18)(14,19)(15,20)(21,28)(22,29)(23,31)(24,32)(25,33)(26,34)(27,30)(35,38)(36,40)(37,39);(9,11,15,14)(10,16,20,19)(12,21,24,27)(13,22,26,25)(17,30,32,28)(18,33,34,29)(23,35,36,37)(31,38,40,39);(1,3,5,4)(2,6,7,8)(9,12,15,24)(10,17,20,32)(11,21,14,27)(13,23,26,36)(16,28,19,30)(18,31,34,40)(22,35,25,37)(29,38,33,39);(1,4,5,3)(2,6,7,8)(9,13)(10,18)(11,22)(12,23)(14,25)(15,26)(16,29)(17,31)(19,33)(20,34)(21,35)(24,36)(27,37)(28,38)(30,39)(32,40);(9,14,15,11)(10,19,20,16)(12,21,24,27)(13,25,26,22)(17,28,32,30)(18,33,34,29)(23,35,36,37)(31,38,40,39)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284740, #72286224, #72286235
maximal normal subgroup of	#72284740, #72286224, #72286235

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6359080);