#6439973

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	#24455	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2153 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi,e^d=ei

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, 6347169);