#6466528

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2022 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)(8,9)(10,12)(13,14)(15,24)(16,29)(17,22)(18,33)(19,23)(20,27)(21,30)(25,39)(26,35)(28,38)(31,44)(32,36)(34,41)(37,40)(42,43);(2,3)(6,9)(10,11)(13,15,17,26,19,28,18,27)(14,20,33,38,23,35,22,24)(16,25,31,41,30,40,32,42)(21,34,44,39,29,43,36,37);(1,2,4,3)(5,7,8,12)(6,10,9,11)(13,16,19,30)(14,21,23,29)(15,25,28,40)(17,31,18,32)(20,34,35,43)(22,36,33,44)(24,37,38,39)(26,41,27,42);(1,3,4,2)(16,30)(21,29)(25,40)(31,32)(34,43)(36,44)(37,39)(41,42);(13,17,19,18)(14,22,23,33)(15,26,28,27)(16,31,30,32)(20,24,35,38)(21,36,29,44)(25,41,40,42)(34,37,43,39)

Polycyclic presentation:

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

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, 6373724);