#6467511

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2252 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)(13,14,18,20,19,24,17,23)(15,21,25,32,27,34,28,36)(16,22,29,33,31,37,30,38)(26,35,39,42,40,43,41,44);(2,4)(14,20)(15,25)(16,29)(17,18)(21,34)(22,37)(23,24)(26,40)(27,28)(30,31)(35,44)(42,43);(6,9)(10,12)(13,15,18,25,19,27,17,28)(14,21,20,32,24,34,23,36)(16,26,29,39,31,40,30,41)(22,35,33,42,37,43,38,44);(13,16,17,30,19,31,18,29)(14,22,23,38,24,37,20,33)(15,26,28,41,27,40,25,39)(21,35,36,44,34,43,32,42);(5,7,8,11)(6,10,9,12)(15,27)(21,34)(25,28)(26,40)(32,36)(35,43)(39,41)(42,44)

Polycyclic presentation:

2^9|a^2=fi,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,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374707);