#6466192

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2184 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)(8,9)(10,12)(13,14,20,24,19,23,18,17)(15,25,29,21,28,36,27,38)(16,30,35,32,33,22,34,40)(26,39,31,44,42,41,43,37);(2,4)(6,9)(10,11)(13,15,19,28)(14,21,23,38)(16,31,33,43)(17,36,24,25)(18,29,20,27)(22,39,30,41)(26,34,42,35)(32,44,40,37);(5,7,8,12)(6,10,9,11)(13,16,18,34,19,33,20,35)(14,22,17,32,23,30,24,40)(15,26,27,43,28,42,29,31)(21,37,25,41,38,44,36,39);(13,17,18,23,19,24,20,14)(15,21,27,25,28,38,29,36)(16,32,34,30,33,40,35,22)(26,37,43,41,42,44,31,39);(16,33)(22,30)(26,42)(31,43)(32,40)(34,35)(37,44)(39,41)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=gh,d^2=hi,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, 6373388);