#6809984

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2072 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)(7,11)(8,10)(9,12)(13,14,17,21)(15,23,25,19)(16,27,29,38)(18,32,31,22)(20,33,37,30)(24,40,28,44)(26,35,42,39)(34,41,36,43);(2,3)(5,7)(6,9)(8,12)(10,11)(13,15,17,25)(14,19,21,23)(16,28,29,24)(18,26,31,42)(20,36,37,34)(22,35,32,39)(27,40,38,44)(30,43,33,41);(1,2,4,3)(7,12)(9,11)(13,16,17,29)(14,20,21,37)(15,24,25,28)(18,33,31,30)(19,34,23,36)(22,38,32,27)(26,43,42,41)(35,40,39,44);(6,10)(9,11)(14,21)(15,25)(20,37)(22,32)(24,28)(26,42)(27,38)(41,43);(1,3,4,2)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717180);