#6606329

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#37682	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2121 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6513525);