#6438247

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3013	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,23)(15,20,27,36)(16,21,30,40)(17,31,18,22)(24,33,26,35)(25,34,42,43)(28,37,41,44)(29,39,32,38);(6,9)(10,12)(13,15,19,27)(14,20,23,36)(16,28,30,41)(17,26,18,24)(21,37,40,44)(22,33,31,35)(25,32,42,29)(34,38,43,39);(2,4)(5,7,8,11)(6,10,9,12)(15,24)(17,18)(20,33)(22,31)(25,28)(26,27)(29,32)(34,37)(35,36)(38,39)(41,42)(43,44);(13,16)(14,21)(15,25)(17,29)(18,32)(19,30)(20,34)(22,38)(23,40)(24,28)(26,41)(27,42)(31,39)(33,37)(35,44)(36,43);(13,17,19,18)(14,22,23,31)(15,26,27,24)(16,29,30,32)(20,33,36,35)(21,38,40,39)(25,41,42,28)(34,37,43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345443);