#6686917

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682380
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	#30502	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1916 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6594113);