#6440414

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3486	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dhi,f^b=fi,h^b=hi,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, 6347610);