#6607255

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×#3303	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2279 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)(15,25,29,36)(16,30,32,40)(17,23,18,22)(20,28,37,27)(21,33,39,34)(26,35,31,38)(41,42,44,43);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,28,19,29,17,27)(14,20,23,36,24,37,22,25)(16,31,34,42,32,26,33,43)(21,38,30,41,39,35,40,44);(7,12)(9,10)(13,16,17,33,19,32,18,34)(14,21,22,40,24,39,23,30)(15,26,27,42,29,31,28,43)(20,35,25,41,37,38,36,44);(5,7)(6,10)(8,12)(9,11)(16,32)(21,39)(26,31)(30,40)(33,34)(35,38)(41,44)(42,43);(7,12)(9,10)(16,32)(21,39)(26,31)(30,40)(33,34)(35,38)(41,44)(42,43)

Polycyclic presentation:

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

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, 6514451);