#6432265

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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	#23421	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2324 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)(15,20)(16,21)(17,23)(18,24)(19,22)(25,33)(26,35)(27,34)(28,36)(29,38)(30,37)(31,39)(32,40)(41,44)(42,43);(6,9)(10,12)(13,15)(14,20)(16,28)(17,26)(18,31)(19,27)(21,36)(22,34)(23,35)(24,39)(25,29)(30,41)(32,42)(33,38)(37,44)(40,43);(2,4)(13,16)(14,21)(15,25)(17,29)(18,32)(19,30)(20,33)(22,37)(23,38)(24,40)(26,41)(27,42)(28,31)(34,43)(35,44)(36,39);(5,7,8,11)(6,10,9,12)(14,22)(20,34)(21,37)(23,24)(33,43)(35,39)(36,44)(38,40);(13,17,19,18)(14,23,22,24)(15,26,27,31)(16,29,30,32)(20,35,34,39)(21,38,37,40)(25,41,42,28)(33,44,43,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339461);