#6810469

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1977 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=fhi,d^2=i,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717665);