#6606990

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3283	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2282 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)(8,11)(9,12)(13,14)(15,25)(16,30)(17,22)(18,23)(19,24)(20,28)(21,32)(26,34)(27,36)(29,35)(31,38)(33,37)(39,40)(41,43)(42,44);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,27,19,29,17,28)(14,20,22,35,24,36,23,25)(16,26,32,40,33,42,31,41)(21,34,30,43,38,44,37,39);(7,12)(9,10)(13,16,18,32,19,33,17,31)(14,21,22,30,24,38,23,37)(15,26,27,40,29,42,28,41)(20,34,35,43,36,44,25,39);(5,7)(6,10)(8,12)(9,11)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,31,33,32)(20,25,36,35)(21,30,38,37)(26,40,42,41)(34,39,44,43);(7,12)(9,10)(13,17,19,18)(14,22,24,23)(15,28,29,27)(16,31,33,32)(20,35,36,25)(21,30,38,37)(26,41,42,40)(34,43,44,39)

Polycyclic presentation:

2^9|b^2=fhi,c^2=g,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^d=eh

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, 6514186);