#6440437

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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×#3489	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4230 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)(13,14,20,26)(15,27,32,38)(16,22,34,42)(17,23,33,41)(18,25,19,24)(21,31,39,30)(28,40,35,37)(29,36,43,44);(2,4)(5,7,8,11)(6,9,10,12)(13,15)(14,21)(16,29)(17,35)(18,31)(19,30)(20,32)(22,40)(23,36)(24,38)(25,27)(26,39)(28,33)(34,43)(37,42)(41,44);(6,10)(9,12)(13,16,20,34)(14,22,26,42)(15,28,32,35)(17,18,33,19)(21,36,39,44)(23,25,41,24)(27,40,38,37)(29,31,43,30);(13,17)(14,23)(15,29)(16,18)(19,34)(20,33)(21,37)(22,25)(24,42)(26,41)(27,36)(28,31)(30,35)(32,43)(38,44)(39,40);(13,18,20,19)(14,24,26,25)(15,30,32,31)(16,33,34,17)(21,27,39,38)(22,23,42,41)(28,29,35,43)(36,40,44,37)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,e^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=dh,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,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, 6347633);