#6449099

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437935
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	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3455 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,17)(13,15)(14,25)(18,24)(19,30)(20,31)(21,28)(22,33)(23,29)(26,32)(27,34)(35,39)(36,38)(37,40);(10,15)(11,19,21,26)(12,22,23,27)(16,30,28,32)(17,33,29,34)(18,25)(31,38)(39,40);(2,5)(6,8)(9,11)(10,16)(12,20)(13,21)(14,26)(15,28)(17,31)(18,30)(19,24)(22,35)(23,36)(25,32)(27,37)(29,38)(33,39)(34,40);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,20)(13,23)(14,27)(15,29)(16,31)(18,33)(19,35)(21,36)(22,24)(25,34)(26,37)(28,38)(30,39)(32,40);(11,21)(16,28)(19,26)(20,36)(30,32)(31,38)(35,37)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318726, #72319082
maximal normal subgroup of	#72318726, #72319082

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356295);