#6445294

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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2049 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:

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

Polycyclic presentation:

2^9|c^2=gi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,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, 6352490);