#6449092

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437931
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
prime radical	#60415	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3340 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)(11,15,20,29)(12,16,23,32)(13,25,27,17)(19,28,35,40)(21,37,36,30)(22,39,38,31)(24,34,26,33);(9,11)(10,15)(12,22,23,38)(13,21)(14,20)(16,31,32,39)(17,30)(18,29)(19,24,35,26)(25,37)(27,36)(28,33,40,34);(2,5)(6,8)(9,12)(10,16)(11,19)(13,26)(14,23)(15,28)(17,34)(18,32)(20,35)(21,38)(22,36)(24,27)(25,33)(29,40)(30,39)(31,37);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,26)(14,23)(15,28)(17,34)(18,32)(20,35)(21,38)(22,36)(24,27)(25,33)(29,40)(30,39)(31,37);(11,20)(15,29)(19,35)(21,36)(22,38)(28,40)(30,37)(31,39)

Polycyclic presentation:

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356288);