#6467560

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2250 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)(7,10)(8,9)(11,12)(13,14,17,22,19,24,18,23)(15,25,27,36,29,35,28,20)(16,21,32,39,31,38,33,40)(26,41,43,44,42,37,30,34);(2,4)(13,15,19,29)(14,20,24,36)(16,30,31,43)(17,28,18,27)(21,37,38,41)(22,35,23,25)(26,32,42,33)(34,39,44,40);(6,9)(10,12)(13,16,18,33,19,31,17,32)(14,21,23,40,24,38,22,39)(15,26,28,30,29,42,27,43)(20,34,35,37,36,44,25,41);(13,16,17,32,19,31,18,33)(14,21,22,39,24,38,23,40)(15,26,27,43,29,42,28,30)(20,34,25,41,36,44,35,37);(5,7,8,11)(6,10,9,12)(16,31)(21,38)(26,42)(30,43)(32,33)(34,44)(37,41)(39,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=hi,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,e^c=ei,e^d=ei

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, 6374756);