#6607085

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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×#3269	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2267 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)(13,14,19,24)(15,25,27,36)(16,30,31,38)(17,22,18,23)(20,28,35,29)(21,32,37,33)(26,34,42,44)(39,41,43,40);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,28,19,27,17,29)(14,20,22,36,24,35,23,25)(16,26,33,40,31,42,32,41)(21,34,30,43,37,44,38,39);(7,12)(9,10)(13,16,17,32,19,31,18,33)(14,21,23,38,24,37,22,30)(15,26,29,41,27,42,28,40)(20,34,25,39,35,44,36,43);(5,7)(6,10)(8,12)(9,11)(15,27)(16,31)(20,35)(21,37)(25,36)(28,29)(30,38)(32,33);(7,12)(9,10)(13,17,19,18)(14,22,24,23)(15,28,27,29)(16,32,31,33)(20,25,35,36)(21,30,37,38)(26,40,42,41)(34,39,44,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=gi,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=ehi,d^c=dhi,e^d=eh

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, 6514281);